Basic Hypergeometric Series (Encyclopedia of Mathematics and its Applications)

ENCYCLOPEDIA

OF MATHEMATICS AND ITS ApPLICATIONS

Editorial Board R. S. Doran, P. Flajolet, M. Ismail, T.-y' Lam, E. Lutwak

Volume 96

Basic Hypergeometric Series Second Edition This revised and expanded new edition will continue to meet the need for an authoritative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. It contains almost all of the important summation and transformation formulas of basic hypergeometric series one needs to know for work in fields such as combinatorics, number theory, modular forms, quantum groups and algebras, probability and statistics, coherent-state theory, orthogonal polynomials, or approximation theory. Simplicity, clarity, deductive proofs, thoughtfully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are devoted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, and generating functions. Chapters 9 to 11 are new for the second edition, the final chapter containing a simplified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Elsewhere some new material and exercises have been added to reflect recent developments, and the bibliography has been revised to maintain its comprehensive nature.

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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

BASIC HYPERGEOMETRIC SERIES

Second Edition

GEORGE GASPER Northwestern University, Evanston, Illinois, USA MIZAN RAHMAN Carleton University, Ottawa, Canada

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc6n 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org

© Cambridge University Press 1990, 2004 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1990 Second edition 2004. Printed in the United Kingdom at the University Press, Cambridge

Typeface Computer Modern 10/12 pt.

System 'lEX

[TB]

A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Gasper, George. Basic hypergeometric series / George Gasper, Mizan Rahman. - 2nd edn. p. cm. - (Encyclopedia of mathematics and its applications; v. 96) Includes bibliographical references and indexes. ISBN 0 521 83357 4 1. Hypergeometric series. I. Rahman, Mizan. II. Title. III. Series. QA353.H9G37 2004 515'.243--dc22 2004045686 ISBN 0 521 83357 4 hardback

The publisher has used its best endeavors to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.

To Brigitta, Karen, and Kenneth Gasper and Babu, Raja, and to the memory of Parul S. Rahman

Contents

page xiii xxi xxv

Foreword Preface Preface to the second edition

1 Basic hypergeometric series

1 1 1

1.1 Introduction 1.2 Hypergeometric and basic hypergeometric series 1.3 The q-binomial theorem 1.4 Heine's transformation formulas for

8 2¢1

series

13

1.5 Heine's q-analogue of Gauss' summation formula

14

1.6 Jacobi's triple product identity, theta functions, and elliptic numbers

15

1. 7 A q-analogue of Saalschiitz's summation formula

17

1.8 The Bailey-Daum summation formula

18

1.9 q-analogues of the Karlsson-Minton summation formulas

18

1.10 The q-gamma and q-beta functions

20

1.11 The q- integral

23

Exercises Notes

24 34

2 Summation, transformation, and expansion formulas

38

2.1 Well-poised, nearly-poised, and very-well-poised hypergeometric and basic hypergeometric series

38

2.2 A general expansion formula

40

2.3 A summation formula for a terminating very-well-poised 4¢3 series

41

2.4 A summation formula for a terminating very-well-poised 6¢5 series

42

2.5 Watson's transformation formula for a terminating very-well-poised 8¢7

series

42

2.6 Jackson's sum of a terminating very-well-poised balanced 8¢7 series

43

2.7 Some special and limiting cases of Jackson's and Watson's formulas: the Rogers-Ramanujan identities

44

2.8 Bailey's transformation formulas for terminating 5¢4 and 7¢6 series

45

2.9 Bailey's transformation formula for a terminating

47

vii

1O¢9

series

viii

Contents

2.10 Limiting cases of Bailey's 1O¢9 transformation formula

48

2.11 Bailey's three-term transformation formula for VWP-balanced S¢7 series

53

2.12 Bailey's four-term transformation formula for balanced 1O¢9 series

55

Exercises Notes

58 67

3 Additional summation, transformation, and expansion formulas

69

3.1 Introduction 3.2 Two-term transformation formulas for 3¢2 series

69 70

3.3 Three-term transformation formulas for 3¢2 series

73

3.4 Transformation formulas for well-poised 3¢2 and very-well-poised S¢4 series with arbitrary arguments

74

3.5 Transformations of series with base q2 to series with base q

77

3.6 Bibasic summation formulas 3.7 Bibasic expansion formulas

80 84

3.8 Quadratic, cubic, and quartic summation and transformation formulas

88

3.9 Multibasic hypergeometric series

95

3.10 Transformations of series with base q to series with base q2 Exercises Notes

4 Basic contour integrals 4.1 Introduction 4.2 Watson's contour integral representation for 2¢1 (a, b; c; q, z) series 4.3 Analytic continuation of

96 100 111

2¢1(a, b; c; q, z)

113 113 115 117

4.4 q-analogues of Barnes' first and second lemmas

119

4.5 Analytic continuation of r+l ¢r series

120

4.6 Contour integrals representing well-poised series

121

4.7 A contour integral analogue of Bailey's summation formula

123

4.8 Extensions to complex q inside the unit disc

124

4.9 Other types of basic contour integrals

125

4.10 General basic contour integral formulas

126

Contents

ix

4.11 Some additional extensions of the beta integral

129

4.12 Sears' transformations of wen-poised series

130

Exercises Notes

132 135

5 Bilateral basic hypergeometric series 5.1 Notations and definitions 5.2 Ramanujan's sum for 1 'l/Jl (a; b; q,

137 137 138

z)

5.3 Bailey's sum of a very-wen-poised 6'l/J6 series

140

5.4 A general transformation formula for an r'l/Jr series

141

5.5 A general transformation formula for a very-wen-poised 2r'l/J2r series

143

5.6 Transformation formulas for very-wen-poised 8'l/J8 and

145

1O'l/J1O

series

Exercises Notes

146 152

6 The Askey-Wilson q- beta integral and some associated formulas

154

6.1 The Askey-Wilson q-extension of the beta integral

154

6.2 Proof of formula (6.1.1)

156

6.3 Integral representations for very-wen-poised 8¢7 series

157

6.4 Integral representations for very-wen-poised

159

1O¢9

series

6.5 A quadratic transformation formula for very-wen-poised balanced series 6.6 The Askey-Wilson integral when max

(Ial, Ibl, lei, Idl)

1O¢9

~ 1

Exercises Notes

7 Applications to orthogonal polynomials

162 163 168 173

175

7.1 Orthogonality

175

7.2 The finite discrete case: the q-Racah polynomials and some special cases

177

7.3 The infinite discrete case: the little and big q-Jacobi polynomials

181

7.4 An absolutely continuous measure: the continuous q-ultraspherical polynomials 7.5 The Askey-Wilson polynomials

184 188

Contents

x

7.6 Connection coefficients 7.7 A difference equation and a Rodrigues-type formula for the

195 197 199

Askey-Wilson polynomials Exercises Notes

213

8 Further applications 8.1 Introduction 8.2 A product formula for balanced

217 4 2,

q

> 1 is probably Pfaff's

(C - a)n(c - b)n (c)n(c - a - b)n'

]

1

n

=

0,1, ....

This result from 1797, see Pfaff [1797], contains as a limit when n ----; important result usually attributed to Gauss [1813],

F [a, b. 1] 2

c'

1

=

r(c)r(c-a-b) r(c-a)r(c-b)'

Re (c-a-b) > O.

00,

(6)

another

(7)

The next instance is a very important result of Clausen [1828]:

{ F [ 2

1

a, b ·x] a + b+ ~ ,

}2 =

3

F [2a, 2b, a + b ·x] 2 a + b + ~, 2a + 2b ' .

(8)

Some of the interest in Clausen's formula is that it changes the square of a class of 2Fl'S to a 3F2. In this direction it is also interesting because it was probably the first instance of anyone finding a differential equation satisfied by [Y(X)]2, y(x)z(x) and [Z(X)]2 when y(x) and z(x) satisfy

a(x)y" + b(x)y' + c(x)y

=

O.

(9)

Foreword

xv

This problem was considered for (9) by Appell, see Watson [1952]' but the essence of his general argument occurs in Clausen's paper. This is a common phenomenon, which is usually not mentioned when the general method is introduced to students, so they do not learn how often general methods come from specific problems or examples. See D. and G. Chudnovsky [1988] for an instance of the use of Clausen's formula, where a result for a 2Fl is carried to a 3F2 and from that to a very interesting set of expansions of 1[-1. Those identities were first discovered by Ramanujan. Here is Ramanujan's most impressive example:

There is another important reason why Clausen's formula is important. It leads to a large class of 3F2 's that are nonnegative for the power series variable between -1 and 1. The most famous use of this is in the final step of de Branges' solution of the Bieberbach conjecture, see de Branges [1985]. The integral of the 2Fl or Jacobi polynomial he had is a 3F2' and its positivity is an easy consequence of Clausen's formula, as Gasper had observed ten years earlier. There are other important results which follow from the positivity in Clausen's identity. Once Kummer [1836] wrote his long and important paper on 2Fl 's and IFI 's, this material became well-known. It has been reworked by others. Riemann redid the 2Fl using his idea that the singularities of a function go a long way toward determining the function. He showed that if the differential equation (9) has regular singularities at three points, and every other point in the extended complex plane is an ordinary point, then the equation is equivalent to the hypergeometric equation x(l - x)y"

+ [c -

(a

+ b + l)x]y' -

aby = 0,

(11)

which has regular singular points at x = 0,1,00. Riemann's work was very influential, so much so that much of the mathematical community that considered hypergeometric functions studied them almost exclusively from the point of view of differential equations. This is clear in Klein's book [1933], and in the work on multiple hypergeometric functions that starts with Appell in 1880 and is summarized in Appell and Kampe de Feriet [1926]. The integral representations associated with the differential equation point of view are similar to Euler's integral representation. This is

[a,

F 2

1

b. x]

c'

=

r(c) (1(1 _ xt)-at b- 1 (1 _ t)c-b- 1 dt r(b)r(c - b) io '

(12)

Ixl < 1, Re c > Re b > 0, and includes related integrals with different contours. The differential equation point of view is very powerful where it works, but it

xvi

Foreword

does not work well for p 2: 3 or q 2: 2 as Kummer discovered. Thus there is a need to develop other methods to study hypergeometric functions. In the late 19th and early 20th century a different type of integral representation was introduced. These two different types of integrals are best represented by Euler's beta integral

r1t

io

a-

1(1 _ t)b-1dt = r(a)r(b) , r(a+b)

Re (a, b)

>0

(13)

and Barnes' beta integral

-1

2n

1

00

r(a + it)r(b + it)r(c - it)r(d - it) dt

(14)

-00

r(a + c)r(a + d)r(b + c)r(b + d) ( ) ra+b+c+d

Re (a, b, c, d)

> o.

There is no direct connection with differential equations for integrals like (14), so it stands a better chance to work for larger values of p and q. While Euler, Gauss, and Riemann and many other great mathematicians wrote important and influential papers on hypergeometric functions, the development of basic hypergeometric functions was much slower. Euler and Gauss did important work on basic hypergeometric functions, but most of Gauss' work was unpublished until after his death and Euler's work was more influential on the development of number theory and elliptic functions. Basic hypergeometric series are series L Cn with cn+I/cn a rational function of qn for a fixed parameter q, which is usually taken to satisfy Iql < 1, but at other times is a power of a prime. In this Foreword Iql < 1 will be assumed. Euler summed three basic hypergeometric series. The one which had the largest impact was 00

2) _1)n q(3n -n)/2 = 2

(q; q)oo,

(15)

-00

where 00

(a; q)oo

=

II (1 -

aqn).

(16)

n=O If

(17) then Euler also showed that

1 xn ..,.....-....,...--2:-(x; q)oo - n=O (q; q)n' 00

Ixl 0, gave its (contiguous) recurrence relations, and derived his famous formula (see (1.2.11) below) for the sum of this series when z = 1 and Re(c-a-b»O. 1

Basic hypergeometric series

2

Although Gauss used the notation F(a, b, c, z) for his series, it is now customary to use F(a, b; c; z) or either of the notations 2Fl

(a, b; c; z),

for this series (and for its sum when it converges), because these notations separate the numerator parameters a, b from the denominator parameter c and the variable z. In view of Gauss' paper, his series is frequently called Gauss'series. However, since the special case a = 1, b = c yields the geometric senes 1+z+z2 +z3 + ... , Gauss' series is also called the (ordinary) hypergeometric series or the Gauss hypergeometric series. Some important functions which can be expressed by means of Gauss' series are (1 + z)a = F( -a, b; b; -z), 10g(1

+ z) = zF(l, 1; 2; -z),

sin- 1 z = zF(1/2, 1/2; 3/2; z2), tan- 1

z

e

Z

=

(1.2.2)

zF(1/2, 1;3/2; _z2),

= a-HXl lim F(a, b; b; z/a),

where Izl < 1 in the first four formulas. Also expressible by means of Gauss' series are the classical orthogonal polynomials, such as the Tchebichef polynomials of the first and second kinds

= F( -n, n; 1/2; (1 - x)/2), = (n + l)F( -n, n + 2; 3/2; (1 - x)/2),

Tn(x) Un(x)

(1.2.3) (1.2.4)

the Legendre polynomials

Pn(x)

=

F( -n, n

+ 1; 1; (1

- x)/2),

(1.2.5)

the Gegenbauer (ultraspherical) polynomials

(2'\)n F( -n, n + 2'\;'\ + 1/2; (1 - x)/2), n. and the more general Jacobi polynomials >.

en (x)

= -,-

p~a,(3)(x)= where n =

(a+,l)n F(-n,n+a+,B+1; a+1;(1-x)/2), n. 0,1, ... , and (a)n denotes the shifted factorial defined by

(a)o=l, (a)n=a(a+1)···(a+n-1)=

r(a+n) r(a) ,

(1.2.6)

(1.2.7)

n=1,2, .... (1.2.8)

Before Gauss, Chu [1303] (see Needham [1959, p. 138], Takacs [1973] and Askey [1975, p. 59]) and Vandermonde [1772] had proved the summation formula (c- b)n (1.2.9) F( -n, b; c; 1) = (c)n ' n = 0,1, ... ,

1.2 Hypergeometric series

3

which is now called Vandermonde's formula or the Chu- Vandermonde formula, and Euler [1748] had derived several results for hypergeometric series, including his transformation formula

F(a, b; c; z)

=

(1 - z)c-a-b F(c - a, c - b; c; z),

Formula (1.2.9) is the terminating case a

F(a, b; c; 1)

=

r(c)r(c - a - b) r(c _ a)r(c _ b)'

Izl < 1.

(1.2.10)

= -n of the summation formula Re(c - a - b) > 0,

(1.2.11)

which Gauss proved in his paper. Thirty-three years after Gauss' paper, Heine [1846, 1847, 1878] introduced the series (1 - qa)(l - qb) (1 - qa)(l - qa+ 1)(1 - qb)(l - qb+ I) 2 1+ (1 _ q)(l _ qc) z+ (1- q)(l _ q2)(1 _ qC)(l _ qC+ I) Z +- .. , (1.2.12) where it is assumed that q i- 1, c i- 0, -1, -2, ... and the principal value of each power of q is taken. This series converges absolutely for Izl < 1 when Iql < 1 and it tends (at least termwise) to Gauss' series as q ----> 1, because 1 _ qa lim - - = a. (1.2.13) q--+I 1 - q The series in (1.2.12) is usually called Heine's series or, in view of the base q, the basic hypergeometric series or q-hypergeometric series. Analogous to Gauss' notation, Heine used the notation ¢(a, b, c, q, z) for his series. However, since one would like to also be able to consider the case when q to the power a, b, or c is replaced by zero, it is now customary to define the basic hypergeometric series by

¢(a,b;c;q,z) == 2¢I(a,b;c;q,z) ==

=

f

~a; q~n~b; q~n

2¢1

[a~b;q,z]

zn,

(1.2.14)

n= 0 q; q n c; q n

where

(a; q)n

= {

~i _ a)(l -

aq) ... (1 _ aqn-I),

n=O, n= 1,2, ... ,

(1.2.15)

is the q-shifted factorial and it is assumed that c i- q-rn for m = 0,1, .... Some other notations that have been used in the literature for the product (a; q)n are (a)q,n' [a]n, and even (a)n when (1.2.8) is not used and the base is not displayed. Another generalization of Gauss' series is the (generalized) hypergeometric series with r numerator parameters aI, ... ,ar and s denominator parameters bl , ... ,bs defined by

rFs(al' a2, ... , ar; bl , ... , bs ;

z) == rFs [alb' a2,·· ·b' ar ; z] I, ... , s

(1.2.16)

Basic hypergeometric series

4

Some well-known special cases are the exponential function (1.2.17)

e Z = oFo(-;-;z),

the trigonometric functions sin z

=

cosz

=

z oFI (-; 3/2; -z2/4), oFI (-; 1/2;

_Z2

/4),

(1.2.18)

the Bessel function Ja(z) = (z/2t oFI (-; a

+ 1; _z2 /4)/r(a + 1),

(1.2.19)

where a dash is used to indicate the absence of either numerator (when r = 0) or denominator (when s = 0) parameters. Some other well-known special cases are the Hermite polynomials Hn(x) = (2x)n 2Fo( -n/2, (1 - n)/2; -; _x- 2 ),

(1.2.20)

and the Laguerre polynomials

= (a +,l)n

(1.2.21 ) IFI ( -n; a + 1; x ) . n. Generalizing Heine's series, we shall define an r¢s basic hypergeometric series by Lna (x)

A. ( A. [aI, a2, ... , a r r'f's al,a2,···,ar ; bI,···, bs;q,z ) -= r'f's b b ;q,z ]

=

f n= 0

1, ... , s

(al;q)n(a2;q)n···(ar ;q)n (q; q)n(bl ; q)n··· (b s; q)n

[(_l t qG)]I+s-r zn

(1.2.22)

with (~) = n(n - 1)/2, where q -=I 0 when r > s + 1. In (1.2.16) and (1.2.22) it is assumed that the parameters bl , ... , bs are such that the denominator factors in the terms of the series are never zero. Since (1.2.23) (-m)n = (q-m; q)n = 0, n = m + 1, m + 2, ... , an rFs series terminates if one of its numerator parameters is zero or a negative integer, and an r¢s series terminates if one of its numerator parameters is of the form q-m with m = 0,1,2, ... , and q -=I O. Basic analogues of the classical orthogonal polynomials will be considered in Chapter 7 as well as in the exercises at the ends of the chapters. Unless stated otherwise, when dealing with nonterminating basic hypergeometric series we shall assume that Iql < 1 and that the parameters and variables are such that the series converges absolutely. Note that if Iql > 1, then we can perform an inversion with respect to the base by setting p = q-I and using the identity (1.2.24) to convert the series (1.2.22) to a similar series in base p with Ipi < 1 (see Ex. 1.4(i)). The inverted series will have a finite radius of convergence if the original series does.

1.2 Hypergeometric series

5

Observe that if we denote the terms of the series (1.2.16) and (1.2.22) which contain zn by Un and Vn , respectively, then U n+ 1

(al

+ n)(a2 + n) ... (a r + n)

--= Un (l+n)(b 1 +n)···(bs+n)

z

(1.2.25)

is a rational function of n, and

(1.2.26)

L::

L::

is a rational function of qn. Conversely, if 0 Un and 0 Vn are power series with Uo = Vo = 1 such that u n+ dUn is a rational function of nand vn+ l/V n is a rational function of qn, then these series are of the forms (1.2.16) and (1.2.22), respectively. By the ratio test, the rFs series converges absolutely for all z if r ::; s, and for Izl < 1 if r = s + 1. By an extension of the ratio test (Bromwich [1959, p. 241]), it converges absolutely for Izl = 1 if r = s + 1 and Re [b 1 + ... + bs (a 1 + ... + ar )] > o. If r > s + 1 and z i=- 0 or r = s + 1 and Iz I > 1, then this series diverges, unless it terminates. If 0 < Iql < 1, the r¢s series converges absolutely for all z if r ::; sand for Izl < 1 if r = s + 1. This series also converges absolutely if Iql > 1 and Izl < Ib 1b2·· ·bsql/lala2·· ·arl· It diverges for z i=- 0 if 0 < Iql < 1 and r > s + 1, and if Iql > 1 and Izl > Ib1 b2·· ·bsql/lala2· ··arl, unless it terminates. As is customary, the rFs and r¢s notations are also used for the sums of these series inside the circle of convergence and for their analytic continuations (called hypergeometric functions and basic hypergeometric functions, respectively) outside the circle of convergence. Observe that the series (1.2.22) has the property that if we replace z by z/a r and let ar ---+ 00, then the resulting series is again of the form (1.2.22) with r replaced by r -1. Because this is not the case for the r¢s series defined n)]l+s-r without the factors [ (_I)nq ( 2 in the books of Bailey [1935] and Slater

[1966] and we wish to be able to handle such limit cases, we have chosen to use the series defined in (1.2.22). There is no loss in generality since the Bailey and Slater series can be obtained from the r = s + 1 case of (1.2.22) by choosing s sufficiently large and setting some of the parameters equal to zero. An r+ 1 Fr series is called k-balanced if b1 + b2 + ... + br = k + al + a2 + ... + ar+ 1 and z = 1; a I-balanced series is called balanced (or Saalschutzian). Analogously, an r+ 1 ¢r series is called k- balanced if b1 b2 ... br = qk al a2 ... ar+ 1 and z = q, and a I-balanced series is called balanced (or Saalschutzian). We will first encounter balanced series in §1.7, where we derive a summation formula for such a series. For negative subscripts, the shifted factorial and the q-shifted factorials are defined by ( a ) -n

1 (a-l)(a-2)···(a-n)

= -:------:--:------:---:------:-

1

(a-n)n

(1.2.27)

6

Basic hypergeometric series

1

1

(a;q)-n= (1-aq-l)(1-aq-2) ... (1-aq-n)

(aq-n;q)n

( -qja)nqG) (qja; q)n ' (1.2.28)

where n

= 0,1, .... We also define 00

(a; q)oo =

II (1- aqk)

(1.2.29)

k= 0

for Iql < 1. Since the infinite product in (1.2.29) diverges when a -=I- 0 and Iql 2:: 1, whenever (a; q)oo appears in a formula, we shall assume that Iql < 1. The following easily verified identities will be frequently used in this book:

() a; q

n =

(

(a; q)oo aq n ; q) 00 ,

(a- 1 ql-n; q)n = (a; q)n( _a- 1 )nq-G), (a;q)n-k =

(a;q)n (_qa-l)kq(~)-nk, (a- 1 ql-n; q)k

(a; q)n+ k = (a; q)n(aqn; q)k,

(1.2.30) (1.2.31 ) (1.2.32) (1.2.33) (1.2.34)

(a;q)n (aq k) ;q n-k = -(-)-, a;q k

(1.2.35)

_ (a; q)n(aqn; q)k (aq 2k.) ,q n-k () , a; q 2k

(1.2.36)

) = (q; q)n (_1)k (~)-nk (q-no,qk () q, q; q n-k

(1.2.37)

q)k(qa- 1 ; q)n -nk (aq -no,q )k -_ (a;(-1 , a q l-k.) ,q n q

(1.2.38)

(a; q)zn

(a; q2)n(aq; q2)n,

(1.2.39)

(a 2; q2)n = (a; q)n( -a; q)n,

(1.2.40)

=

where nand k are integers. A more complete list of useful identities is given in Appendix I at the end of the book. Since products of q-shifted factorials occur so often, to simplify them we shall frequently use the more compact notations

(ai, a2,···, am; q)n = (al; q)n(a2; q)n··· (am; q)n,

(1.2.41 )

(ai, a2, ... , am; q)oo = (al; q)oo (a2; q)oo ... (am; q)oo.

(1.2.42)

1.2 Hypergeometric series

7

The ratio (1 - qa)/(1 - q) considered in (1.2.13) is called a q-number (or basic number) and it is denoted by 1 _ qa (1.2.43) [a]q = - 1 - ' q -I- 1. -q It is also called a q-analogue, q-deformation, q-extension, or a q-generalization of the complex number a. In terms of q-numbers the q-number factorial [n]q! is defined for a nonnegative integer n by n

[n]q! =

II [k]q,

(1.2.44)

k= I

and the corresponding q-number shifted factorial is defined by

n-I [a]q;n

=

II [a + k]q.

(1.2.45)

k=O

Clearly,

· []' 11m n q. -- n.,,

q-tl

lim [a]q q-tl

=

a,

(1.2.46)

and lim [a]q;n = (a)n. q-tl

(1.2.4 7)

Corresponding to (1.2.41) we can use the compact notation

[aI, a2,···, am]q;n = [adq;n[a2]q;n'" [am]q;n. Since

(1.2.48)

f

[al,a2, ... ,ar]q;n [(_I)n qG)]I+s-r zn n= 0 [n]q![b l , ... , bs]q;n

(a1

A.. -_ r'f/s q , qa2 , ... , qar ,• qb, , ... , qb s ,• q, z (1 - q)I+s-r) ,

(1.2.49)

anyone working with q-numbers and the q-number hypergeometric series on the left-hand side of (1.2.49) can use the formulas for reps series in this book that have no zero parameters by replacing the parameters by q"lj powers and applying (1.2.49). As in Frenkel and Turaev [1995] one can define a trigonometric number [a; u] by [a'u] = sin(7rua) (1.2.50) , sin( 7ru) for noninteger values of u and view [a; u] as a trigonometric deformation of a since lima-to [a; u] = a. The corresponding rts trigonometric hypergeometric series can be defined by

rts(al, a2, ... , ar; bl , ... , bs ; u, z)

=

f

[al,a2, ... ,ar ;u]n n= 0 [n; u]![bl , ... , bs; u]n

[(_lt e11"ia G)]I+s-r zn ,

(1.2.51 )

Basic hypergeometric series

8 where

n

[n; u]! =

II [k; u],

n-I

[a; u]n =

k= I

II [a + k; u],

(1.2.52)

k= 0

and (1.2.53) From eniaa _ e- niaa

[a·, u] = -:-----,---e7ria _ e- 7ricr

qa/2 _ q-a/2 _ 1 - qa (l-a)/2 ql/2 _ q-I/2 - 1 - q q ,

(1.2.54)

where q = e 2nia , it follows that

(qa;q)n n(l-a)/2-n(n-I)/4 [a,. u ]n -_ (1 _ q)n q ,

(1.2.55)

rts(al' a2,···, ar ; bl ,···, bs; u, z) rI-. (a1 -_ r'l's q ,qa2 , ... , qar •,q b1 , ... , qbs •,q, cz )

(1.2.56)

and hence

with (1.2.57) which shows that the rts series is equivalent to the reps series in (1.2.49). Elliptic numbers [a; u, T], which are a one-parameter generalization (deformation) of trigonometric numbers, are considered in §1.6, and the corresponding elliptic (and theta) hypergeometric series and their summation and transformation formulas are considered in Chapter 11. We close this section with two identities involving ordinary binomial coefficients, which are particularly useful in handling some powers of q that arise in the derivations of many formulas containing q-series: (1.2.58) (1.2.59)

1.3 The q-binomial theorem One of the most important summation formulas for hypergeometric series is given by the binomial theorem:

2FI (a, C; C; z) where Izl

=

IFo(a;-; z)

=

f

(a),n zn n=O n.

= (1- z)-a,

(1.3.1)

< 1. We shall show that this formula has the following q-analogue

rI-. ( . _ . 1'l'0 a, ,q,

) _

z -

~ (a;q)n n _ (az;q)oo ~ ( ) z - ( ) , n= 0 q; q n Z; q 00

Izl

< 1, Iql < 1,

(1.3.2)

1.3 The q-binomial theorem

9

which was derived by Cauchy [1843]' Heine [1847] and by other mathematicians. See Askey [1980a], which also cites the books by Rothe [1811] and Schweins [1820]' and the remark on p. 491 of Andrews, Askey, and Roy [1999] concerning the terminating form of the q-binomial theorem in Rothe [1811]. Heine's proof of (1.3.2), which can also be found in the books Heine [1878], Bailey [1935, p. 66] and Slater [1966, p. 92]' is better understood if one first follows Askey's [1980a] approach of evaluating the sum of the binomial series in (1.3.1), and then carries out the analogous steps for the series in (1.3.2). Let us set .. ( ) Ja Z

=

~ (a)n n ~ ,z. n=O n.

(1.3.3)

Since this series is uniformly convergent in Izl ::; differentiate it termwise to get

f =f

f~(z) =

E

when 0



, m

(1.3.33) which, because of the normalizing factor that he introduced, has the nice property that Eq(O;a) = 1 (see Suslov [2003, p.17]).

1.4 Heine's transformation formulas for 2 ues of m and n, so that the denominator f)l (1[CT, e7riT) in (1.6.14) is never zero. Then, from (1.6.9) it is clear that [a; CT, T] is well-defined, [-a; CT, T] = -[a;CT,TJ, [1;CT,T] = 1, and . sin( 1[CTa) (1.6.15) hm [a; CT, T] = . ( ) = [a;CT]. III T-tOO sIn 1[CT

°

Hence, the elliptic number [a; CT, T] is a one-parameter deformation of the trigonometric number [a; CT] and a two-parameter deformation of the number a. Notice that [a; CT, T] is called an "elliptic number" even though it is not an elliptic (doubly periodic and meromorphic) function of a. However, [a;CT,T] is a quotient of f)l functions and, as is well-known (see Whittaker and Watson [1965, §21.5]), any (doubly periodic meromorphic) elliptic function can be written as a constant multiple of a quotient of products of f)l functions. The corresponding elliptic hypergeometric series are considered in Chapter 11.

1. 7 A q-analogue of Saalschiitz's summation formula

Pfaff [1797] discovered the summation formula 3 F 2(a,b,-n;c,l+a+b-c-n;I)=

(c - a)n(c - b)n (c)n(c-a-b)n'

n=O,I, ... , (1.7.1)

which sums a terminating balanced 3F2(1) series with argument 1. It was rediscovered by Saalschutz [1890] and is usually called Saalschiitz formula or the Pfaff-Saalschiitz formula; see Askey [1975]. To derive a q-analogue of (1.7.1), observe that since, by (1.3.2),

(abzjc; q)oo (z;q)oo

=

f

(abjc; q)k zk k=O (q;q)k

the right side of (1.4.3) equals

ff

k= 0 m= 0

(abjc; q)k(cja, cjb; q)m (ab)m zk+ m, (q; qh(q, c; q)m C

and hence, equating the coefficients of zn on both sides of (1.4.3) we get

t .

J= 0

(q-n, cja,cjb; q)j qj (q,c,cql-njab;q)j

= (a,b;q)n . (c,abjc;q)n

Replacing a, b by cj a, cjb, respectively, this gives the following sum of a terminating balanced 3 (P2 series

3(P2 (a,b,q -n ;c,abc-I qI-n ;q,q ) =

(cja, cjb; q)n ( j b ) , c, c a ; q n

n=O,I, ... ,

(1. 7.2)

18

Basic hypergeometric series

which was first derived by Jackson [1910a]. It is easy to see that (1.7.1) follows from (1.7.2) by replacing a, b, c in (1. 7.2) by qa, qb, qC, respectively, and letting q -+ 1. Note that letting a -+ 00 in (1.7.2) gives (1.5.2), while letting a -+ 0 gives (1.5.3). 1.8 The Bailey-Daum summation formula

Bailey [1941] and Daum [1942] independently discovered the summation formula ,/, ( b' jb' _ jb) = (-q; q)oc,(aq, aq2 jb2; q2)00 (1.8.1) 2'f'1 a, ,aq ,q, q (aq jb , _ q jb',q) 00 , which is a q-analogue of Kummer's formula

2FI(a,b;l+a-b;-I)=

r(1 (

r

+ a - b)r(1 + ta) ) ( I ). r 1 + 'ia - b

1+a

(1.8.2)

Formula (1.8.1) can be easily obtained from (1.4.1) by using the identity (1.2.40) and a limiting form of (1.2.39), namely, (a; q)oo = (a, aq; q2)00, to see that 2¢1 (a, b; aqjb; q, -qjb)

(a, -q; q)oo ,/, (jb jb ) (aq,q,qoo j b _ jb') 2'f'1 q ,-q ;-q;q,a (a,-q;q)oo ~ (q2jb 2;q2)n n = (aqjb, -qjb; q)oo ~ (q2; q2)n a (a, -q; q)oo (aq2 jb2; q2)00 (aqjb, -qjb; q)oo (a; q2)00 2 (-q; q)oo (aq, aq2 jb ; q2 )00 (aqjb, -qjb; q)oo

by (1.3.2)

1.9 q-analogues of the Karlsson-Minton summation formulas

Minton [1970] showed that if a is a negative integer and ml, m2, ... , mr are nonnegative integers such that -a 2: ml + ... + mT) then

F r+ 2 r+

[a, b, I

bb ++ 1, b l

+ mr . 1]

ml , ... , br l , ... , br

'

r(b + 1)r(1 - a) (b l - b)ml ... (b r - b)m r (1.9.1) r(1 + b - a) (bdml ... (br)m r where, as usual, it is assumed that none of the factors in the denominators of the terms of the series is zero. Karlsson [1971] showed that (1.9.1) also holds when a is not a negative integer provided that the series converges, i.e., if Re( -a) > ml + ... + mr - 1, and he deduced from (1.9.1) that F [a, r+1 r

b+b l

ml , ... , br b 1, ... , r

+ mr , . 1] - 0 - ,

Re (-a)

> ml + ... + mT) (1.9.2)

1.9 q-Karlsson-Minton formulas

19

(1.9.3) These formulas are particularly useful for evaluating sums that appear as solutions to some problems in theoretical physics such as the Racah coefficients. They were also used by Gasper [1981b] to prove the orthogonality on (0,27f) of certain functions that arose in Greiner's [1980] work on spherical harmonics on the Heisenberg group. Here we shall present Gasper's [1981a] derivation of q-analogues of the above formulas. Some of the formulas derived below will be used in Chapter 7 to prove the orthogonality relation for the continuous q- ultraspherical polynomials. Observe that if m and n are nonnegative integers with m ~ n, then

'" (-n -m b ) 2'1-'1 q , q ; r; q, q

=

(brqm; q)n -mn (b') q r,qn

by (1.5.3), and hence

'" [al, ... ,ar,brqm. ] r+ 1 'l-'r b q, Z I,···, br-I, b' r

=

~ ~

n= 0 =

(al, ... ,ar;q)n n~ (q-n,q-m;q)k mn+k Z ~ q (q,bl , ... ,br-I;q)n (q,br;q)k k= 0

ff

n= 0 k= 0

(al, ... ,ar;q)n(q-m;qh zn(_l)kqmn+k-nk+(~) (bl, ... ,br-I;q)n(q;q)n-k(q,br;q)k

~ (q-m,al, ... ,ar;qh( m)k _(k) -zq q 2 k=O (q,bl, ... ,br;q)k

= ~

X

alqk, ... ,arqk m-k] r¢r-I [ b k b k;q,zq , Iq , ... , r-Iq

Izl < 1. (1.9.4)

This expansion formula is a q-analogue of a formula in Fox [1927, (1.11)] and independently derived by Minton [1970, (4)]. When r = 2, formulas (1.9.4), (1.5.1) and (1.5.3) yield

¢ [a,b,b1qm.

3 2

bq,b l

,q,a

-I I-m] _ q

-m

(q,bq/a;q)oo ¢ ( - (bq,q/a;q)oo 2 1 q

b'b' ) "I,q,q

= (q,bq/a;q)oo(bl/b;q)mb m , ( ) ( ) / bq, q a; q

00

bl ; q

(

1.9.5

)

m

provided that la-1ql-ml < 1. By induction it follows from (1.9.4) and (1.9.5) that if ml,"" mr are nonnegative integers and la-1ql-(m 1 +... + mr ) I < 1, then

'" [a,b,blqml, ... ,brqmr. -I 1-(m1+ ... +m r )] r+ 2 'l-'r+ 1 b b b ' q, a q q, 1,···, r = (q, bq/a; q)oo (bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bq, q/a; q)oo (b l ; q)ml ... (b r ; q)mr

(1.9.6)

20

Basic hypergeometric series

which is a q-analogue of (1.9.1). Formula (1.9.1) can be derived from (1.9.6) by replacing a, b, bl , ... , br by qa, qb, qb 1, ... , qb r , respectively, and letting q ----7 1. Setting br = b, mr = 1 and then replacing r by r + 1 in (1.9.6) gives '" [a,blqml, ... ,brqmr. -I -cm1+ ... +mr)] _ 0 I -I -Cm1+···+mr)1 r+ I '/'r bI, ... , br ' q, a q - , a q

< 1,

(1.9.7) while letting b ----700 in the case a '"

r+I'/'r

=

q-Cml+ ... + m r ) of (1.9.6) gives

] [q -Cm1+ ...+mr) ' bI qml , ... , br qmr 1 bI, ... , br ;q,

(_I)ml+···+mr(. ) ( ) q(q m)+"'+mrq-cml+"'+mrHml+"'+mr+I)/2, (1.9.8) bl ; q ml ... br ; q mr

=

which are q-analogues of (1.9.2) and (1.9.3). Another q-analogue of (1.9.3) can be derived by letting b ----7 0 in (1.9.6) to obtain

(1.9.9) when la-Iql-Cm1+···+mr)1 < 1. In addition, if a = q-n and n is a nonnegative integer then we can reverse the order of summation of the series in (1.9.6), (1.9.7) and (1.9.9) to obtain

n 2 ml '" r+ I '/'r

[

+ ... + mn

mr. ] - 0 q -n , bI qml , ... , br q bI, ... , br ' q, q - ,

(1.9.10)

(1.9.11)

and the following generalization of (1.9.8) '" [q-n,blqml, ... ,brqmr. r+ I '/'r bI,···, br ' q,

1] -_ (_I)n(q;q)nq-nCn+I)/2 (b) (b) , I;qml'"

r;qm r

(1.9.12)

where n 2 ml + ... + mn which also follows by letting b ----7 00 in (1.9.10). Note that the b ----7 0 limit case of (1.9.10) is (1.9.11) when n > ml + ... + mn and it is the a = q-Cml+ ... + m r ) special case of (1.9.9) when n = ml + ... +mr .

1.10 The q-gamma and q-beta functions The q-gamma function

r q (x ) = (x.) (q; q)oo q ,q

00

(1 - q )I-x , 0

< q < 1,

(1.10.1)

1.10 The q-gamma and q- beta functions

21

was introduced by Thomae [1869] and later by Jackson [1904e]. Heine [1847] gave an equivalent definition, but without the factor (l-q)l-x. When x = n+ 1 with n a nonnegative integer, this definition reduces to (1.10.2) which clearly approaches n! as q ---+ 1-. Hence f q(n + 1) tends to f( n + 1) = n! as q ---+ 1-. The definition of f q (x) can be extended to Iq I < 1 by using the principal values of qX and (1 - q)l-x in (1.10.1). To show that lim fq(x) = f(x) (1.10.3) q---+I we shall give a simple, formal proof due to Gosper; see Andrews [1986]. From (1.10.1),

Hence lim fq(x q---+I-

+ 1)

=

(+I)X II _n_ _n_ n+x n 00

n= I

=

xr(x)

=

r(x

+ 1)

by Euler's product formula (see Whittaker and Watson [1965, §12.11]) and the well-known functional equation for the gamma function r(x

+ 1) =

xr(x),

r(1) = 1.

(1.10.4)

For a rigorous justification of the above steps see Koornwinder [1990]. From (1.10.1) it is easily seen that, analogous to (1.10.4), fq(x) satisfies the functional equation 1- qX (1.10.5) f(x + 1) = - 1 - f (x), f(l) = 1. -q Askey [1978] derived analogues of many of the well-known properties of the gamma function, including its log-convexity (see the exercises at the end of this chapter), which show that (1.10.1) is a natural q-analogue of f(x). It is obvious from (1.10.1) that fq(x) has poles at x = 0, -1, -2, .... The residue at x = -n is lim (x x---+-n

+ n) f q(x)

(1 _q)n+1 x+n lim (1 - q-n)(1 - ql-n) ... (1 - q-I) x---+-n 1 _ qX+ n (1 _ q)n+ I (1.10.6)

= --;--------:-----:----'---:--'----,-----------,----:-:-

22

Basic hypergeometric series

The q-gamma function has no zeros, so its reciprocal is an entire function with zeros at x = 0, -1, -2, .... Since 1

00

r (x) =

(1- q)x-l

1- qn+ x

II 1 _ qn+

1'

(1.10.7)

n= 0

q

the function 1/r q (x) has zeros at x = -n ± 27rik / log q, where k and n are nonnegative integers. A q-analogue of Legendre's duplication formula r(2x)r

(~)

= 22x - 1r(x)r

(x +~)

(1.10.8)

can be easily derived by observing that

(q,q2;q2)00 (1_ q2)1-2x (q2x , q2x+ 1 ; q2 ) 00

rq2(X)rq2(X+t) r q2 (t) =

(q; q)oo (1 _ q2)1-2x (q2x;q)00

=

(1

+ q)I- 2x r

(2x) q

and hence (1.10.9) Similarly, it can be shown that the Gauss multiplication formula r(nx)(27r)(n-l)/2

=

nnx-!r(x)r (x

+ ~) ... r

(x

+ n: 1 )

(1.10.10)

has a q-analogue of the form

rq(nx)rr

(~) rr (~) ... rr (n: 1)

= (1 + q + ... + qn-l tx-1rr(x)rr (x +

~) ... rr (x + n: 1)

(1.10.11)

with r = qn; see Jackson [1904e, 1905d]. The q-gamma function for q > 1 is considered in Exercise 1.23. For other interesting properties of the q-gamma function see Askey [1978] and Moak [1980a,b] and Ismail, Lorch and Muldoon [1986]. Since the beta function is defined by

B(

X,y

)

=

r(x)r(y)

r(x+y)'

(1.10.12)

it is natural to define the q-beta function by (1.10.13)

1.11 The q-integral which tends to B(x, y) as q

23

1-. By (1.10.1) and (1.3.2),

----+

B (x, y) = (1 _ q) (q, qX+ Y; q)oo

q

(qX,qY;q)oo

= (1 - q) (q; q)oo

f

(qY; q)n qnx (qY;q)oo n=O (q;q)n n+ 1

)

= (1 - q) L (q n+ Y: q) qnx, 00

(

Re x, Re y

00

n=O q

,qoo

> O. (1.10.14)

This series expansion will be used in the next section to derive a q-integral representation for Bq(x, y).

1.11 The q-integral Thomae [1869, 1870] and Jackson [1910c, 1951] introduced the q-integral

11

f(t) dqt = (1 - q) ~ f(qn)qn

(1.11.1)

and Jackson gave the more general definition

lb

f(t) dqt

=

1b

f(t) dqt -

1 a

f(t) dqt,

(1.11.2)

where (1.11.3) Jackson also defined an integral on (0, (0) by (1.11.4) The bilateral q-integral is defined by

1

00

f(t) dqt = (1 - q)

f n=

-00

[f(qn) + f( _qn)] qn.

(1.11.5)

-00

If f is continuous on [0, a], then it is easily seen that

lim q---+l

r f(t) dqt = iofa f(t) dt

(1.11.6)

io

and that a similar limit holds for (1.11.4) and (1.11.5) when restricted. By (1.11.1), it follows from (1.10.14) that

1

f is suitably

t x-I ((tq; q)oo ) dqt, R ex> 0 , y r--I- 0 ,- 1,- 2, ... , tqY;qoo which clearly approaches the beta function integral B q(x, y ) =

1

o

B(x, y)

=

11

t x - 1 (1 - t)y-l dt,

Re x, Re y

> 0,

(1117) ..

(1.11.8)

24

Basic hypergeometric series

as q ----+ 1-. Thomae [1869] rewrote Heine's formula (1.4.1) in the q-integral form 2¢1

) _ fq(c) ( a b. c. q ,q ,q ,q,z - fq(b)fq(c-b)

t t b-I (tz,tqc-b;q)cx; (tzqa,tq;q)cx; dqt,

io

(1.11.9)

which is a q-analogue of Euler's integral representation 2FI

(a, b; c; z)

=

f(b)~~~ _ b)

11

t b- I (1 -

W- b- I (1 -

tz)-a dt,

(1.11.10)

where Iarg(l- z)1 < 7r and Re c> Re b > O. The q-integral notation is, as we shall see later, quite useful in simplifying and manipulating various formulas involving sums of series.

Exercises

1.1 Verify the identities (1.2.30)-(1.2.40), and show that

(aq-n; q)n

(i) (ii)

(iii)

(vi)

(qja; q)n ( _~) n q-G),

(_~)n -nk-(~) (aq -k-n.,q)n = (qja;q)n+k ( , qk q q q ja;) (qa!, -qa!; q)n 1 - aq2n (a!, -a!; q)n 1- a ' (a;q)zn

(iv)

(v)

=

=

(a!,-a!,(aq)!,-(aq)!;q)n'

(a; q)n(qja; q)-n = (-atq(~), (q, -q, _q2; q2)cx; = 1.

1.2 The q-binomial coefficient is defined by

[n]

(q;q)n

k q - (q; q)k(q; q)n-k

for k

= 0,1, ... , n,

and by

(q!3+ I, q-!3+ I; q)cx; (q, q+ I; q)cx; for complex a and f3 when Iql

(i) (ii) (iii)

(q+ I; q)k (q;q)k '

< 1. Verify that

25

Exercises (iv)

(v)

[ -kG

[G:

L[

1

G

+~- 1

L= [~L

qk

L

(_q-a)kq-(~),

lL = [~L + [k ~ lL

+ [k ~

q+ I-k,

k

n

(Z;q)n=2::[~] (-z)k q(2),

(vi)

k= 0 q when k and n are nonnegative integers. 1.3 (i) Show that the binomial theorem

(a+bt

=

t (~)akbn-k k= 0

where n

= 0,1, ... , has a q-analogue of the form (ab; q)n

=

t [~] t [~] k= 0

=

bk(a; q)k(b; q)n-k q

an-k(a; qh(b; q)n-k. k= 0 q (ii) Extend the above formula to the q-multinomial theorem (ala2 ... am+I; q)n

2:: 0:Sklo···,O::;km,

[k

n l , ...

,km

a~l a~l + k2 ... a~: ~2+ ... + krn

] q

kl +···+krn:Sn

x (al; q)k 1 (a2; q)k 2 ... (am; q)krn (a m+I; q)n-(k 1 +... + krn ), where m = 1,2, ... , n = 0,1, ... , and

is the q-multinomial coefficient. 1.4 (i) Prove the inversion formula

r¢s [a l ' ... ' ar ; q, z] bl

, ...

,bs

(-I -I -I) aI' ... ,ar ; q n I ~ ( -I bb- I -I ) n n=oq 'I,···,s;q

= '"""' (Xl

(ii) By reversing the order of summation, show that A.

r+I'Ps

[al' ... ' an q-n . ] b1, ... , bs ,q,z

26

Basic hypergeometric series when n = 0,1, .... (iii) Show that

r+ 1'. q, qz] " [a1'b ... ' ab

r+ 1 'Pr

1, ... , r

t

(al, ... , ar+ 1; q)oo

z-1

= (l-q)(q,b 1 , •.• ,br ;q)00}o t

°

when < q terminate.

(qt, b1t, ... , brt; q)oo d (al t , ... ,ar+l t ;q)oo qt,

< 1, Re z > 0, and the series on the left side does not

1.5 Show that

(e,bqn;q)rn _ (b/e;q)n ~ (q-n,e;q)kqk ( k. ) ~0 ( q, eq 1 -n / b; q) k cq, q rn· ( b; ) q rn - ( b; q) n k= 1.6 Prove the summation formulas

(~:;.q2inq-(~),

(i) 2¢I(q-n,ql-n;qb2;q2,q2) =

,q

n

" (a; e; q, e /) (e/a; )q)oo , (11.. ) 1 'PI a = ( e; q 00 (iii) 2¢o(a, q-n; -; q, qn fa) = a- n , (iv)

2

n 2 -n

00

L - ( q)2 n= q; q n

=

(

)

,

q; q 00 (zp-l ;p-l)oo -1 (v) l¢o(a;-;p,z) = (azp -1 ;p-1) 00 , Ipl > 1, lazp 1 1. e, a e,p 0

1.7 Show that, for

Izl < 1, 2 ) ( ) 2¢1 ( a ,aq;a;q,z = l+az

1.8 Show that, when 2¢1

=

lal < 1 and

(a 2qz; q)oo ( ) . z;q 00

Ibq/a 2 < 1, 1

(a 2, a2/b; b; l, bq/a2)

(a2,q;f)~

[(b/~;q)oo + (-b/~;q)oo] .

2(b, bq/ a ,q)oo

(a, q)oo

(-a, q)oo

(Andrews and Askey [1977]) 1.9 Let ¢(a, b, c) denote the series q-contiguous relations: .

(1) ¢(a, b, eq

-I

2¢1

(a, b; e; q, z). Verify Heine's [1847]

(l-a)(I-b) ) - ¢(a, b, c) = ez ( )( ) ¢(aq, bq, cq), q-e l-e

(ii) ¢(aq, b, c) - ¢(a, b, c) = az ~

=~ ¢(aq, bq, eq),

Exercises

27

... (l-b)(l-c/a) 2 (m) ¢(aq, b, cq) - ¢(a, b, c) = az ( )( ) ¢(aq, bq, cq ), 1-c 1-cq . 1 (1- b/aq) (IV) ¢(aq, bq- ,c) - ¢(a, b, c) = az 1 _ c ¢(aq, b, cq). 1.10 Denoting 2¢1 (a, b; c; q, z), 2¢1 (aq±l, b; c, q, z), 2¢1 (a, bq±l; c; q, z) and 2¢1 (a, b; cq±l; q, z) by ¢, ¢(aq±I), ¢(bq±l) and ¢(cq±I), respectively, show that

(i) b(l - a)¢(aq) - a(l - b)¢(bq)

=

(b - a)¢,

(ii) a(l - b/c)¢(bq-l) - b(l - a/c)¢(aq-l) = (a - b)(l - abz/cq)¢, (iii) q(l - a/c)¢(aq-l) + (1 - a)(l - abz/c)¢(aq) = [1 + q - a - aq/c + a2z(l - b/a)/c]¢, (iv) (1 - c)(q - c) (abz - c)¢(cq-l) + (c - a)(c - b)z¢(cq)

= (c - l)[c(q - c) + (ca + cb - ab - abq)z]¢. (Heine [1847]) 1.11 Let g(e;A,/-l,v) = (AeiO,/-lV;q)oo 2¢1(/-le- iO ,ve- iO ;/-lv;q, Ae iO ). Prove that g(e; A, /-l, v) is symmetric in A, /-l, v and is even in e. 1.12 Let Vq be the q-derivative operator defined for fixed q by

V J(z) = J(z) - J(qz) q (1 - q)z ' and let V~u = Vq(D~-IU) for n = 1,2, .... Show that (i) lim VqJ(z) = dd J(z) if J is differentiable at z,

q---+l (11.. ) 'T"In Vq

z

rI-.

2'/-'1

(b a, ;c;q,z ) =

... ) 'T"In { (z; q)oo (m Vq (abz/c;q)oo

rI-.

2'/-'1

(

(a, b; q)n ) (

c; q

)

1- q

n

rI-. (

n

2 '/-'1

aq n , bqn ;cqn ;q,z ),

(b )} a, ;c;q,z

_ (c/a,c/b;q)n (ab)n (zqn;q)oo ¢ ( b' n. n) - (c; q)n(1- q)n C (abz/c; q)oo 2 1 a, ,cq ,q, zq . (iv) Prove the q-Leibniz formula

V;[J(z)g(z)]

=

~ [~] qV;-k J(zqk)V;g(z).

1.13 Show that u(z) = 2¢1 (a, b; c; q, z) satisfies (for Izl < 1 and in the formal power series sense) the second order q-differential equation

z(c _ abqz)V2 u + [1 - c q 1-q (l-a)(l-b) (1 _ q)2 u = 0,

+ (1 -

a)(l - b) - (1 - abq) z] V u 1-q q

where Vq is defined as in Ex. 1.12. By replacing a, b, c, respectively, by qa, qb, qC and then letting q ----+ 1- show that the above equation tends to the second order differential equation

z(l - z)v" + [c - (a + b + l)z]v' - abv = 0

28

Basic hypergeometric series for the hypergeometric function v(z) (Heine [1847])

1.14 Let Ixl

=

2F1 (a, b; c; z), where Izl < 1.

< 1 and let eq(x) and Eq(x) be as defined in §1.3. Define . Slllq (x)

COS q

=

eq(ix) - eq( -ix) 2. ~

(_I)nx2n+ 1

L (.q, q)2n+ , n= 00

=

0

1

( ) _ eq(ix) +eq(-ix) _ ~ (_I)nx2n x 2 - ~ (.) . n= 0 q, q 2n

Also define

S.

()

lllq

x =

Eq(ix) - Eq( -ix) 2i

'

Show that

eq(ix) = cosq(x) + i sinq(x), Eq(ix) = Cosq(x) + iSinq(x), sinq(x)Sinq(x) + cosq(x)Cosq(x) = 1, sinq(x)Cosq(x) - Sinq(x) cosq(x) = 0.

(i) (ii) (iii) (iv)

For these identities and other identities involving q-analogues of sin x and cosx, see Jackson [1904a] and Hahn [1949c]. 1.15 Prove the transformation formulas A.

(i)

2'/'1

A.

(ii)

2'/'1

[q-n,b. C'

q, Z

] _ (bzq-n/c;q)oo (bz/c; q)oo

[q-n,c/b,o. ] c, cq/bz ,q, q ,

[q-n,b. ] _ (c/b;q)nbn A. [q-n,b,q/z. /] C ,q,z - (C;q)n 3'/'1 bq1-n/C ,q,z C

q-n,b. ] _ (c/b;q)n 21;1 [ C ,q,z - (.) C, q n

(iii)

A.

3 '/'2

31;2

,

°

[q-n,b,bzq-n/C. ] bq 1-n/ C, ,q,q .

(See Jackson [1905a, 1927]) 1.16 Show that

1.17 Show that

t

(a, b; q)k (_ab)n-kq(n-k)(n+ k-1)/2 k=O (q;q)k n

=

(-b)kq(~)

(a; q)n+ 1 ~ (q; qh(q; q)n-k(1 _ aqn-k)·

(Carlitz [1974])

Exercises

29

1.18 Show that

(i) (ii) (iii)

(c;q)oo 1¢I(a;c;q,z) = (z;q)oo 1¢I(az/c;z;q,c), and deduce that 1¢1(-bq;O;q,-q) = (-bq2;q2)00/(q;q2)00, (z;q)oo 2¢I(a,0;c;q,z) = (az;q)oo 1¢2(a;c,az;q,cz),

f

n= 0

((a;2~)n) q, a ,q

qG)(at/z)n 2¢I(q-n,a;ql-n/a;q, qz2/a) n

Iztl < 1.

= (-zt;q)oo 2¢I(a,a/z2;a2;q,-zt), 1.19 Using (1.5.4) show that

(ii)

[a, q/a. -b] = (ab, bq/a; q2)00 -q, b ,q, (b; q)oo ' a2,b2 ] (a 2q,b2q;q2)00 [ b 1 b 1 ; q, -q = a q2,-a q2 (q,a 2b2q;q2)00 . A..

(i)

2'1-'2

2¢2

(Andrews [1973]) 1.20 Prove that if Re x

> 0 and 0 < q < 1, then

(i) (ii)

1.21 For 0

< q < 1 and x > 0, show that d2 2 00 qn+ x dx2 logfq(x) = (logq) ~ (1- qn+ x)2'

which proves that log f q (x) is convex for x

> 0 when 0 < q < 1.

1.22 Conversely, prove that if f(x) is a positive function defined on (0, oo)which satisfies 1- qX f(x + 1) = --f(x) for some q, 0 < q < 1, l-q

f(l) = 1, and logf(x) is convex for x > 0, then f(x) = fq(x). This is Askey's [1978] q-analogue of the Bohr-Mollerup [1922] theorem for r(x). For two extensions to the q > 1 case (with f q (x) defined as in the next exercise), see Moak [1980b]. 1.23 For q > 1 the q-gamma function is defined by f ()= (q-I;q-I)oo( _1)I-X x(x-I)/2

qX

(q-x;q-I)oo q q .

Basic hypergeometric series

30

Show that this function also satisfies the functional equation (1.10.5) and that rq(x) ----t r(x) as q ----t 1+. Show that for q > 1 the residue of rq(x) at x = -n is

n+ 1) (q _ l)n+ 1q ( 2 (q; q)n logq

1.24 Jackson [1905a,b,el gave the following q-analogues of Bessel functions:

(qv+ 1. q) ( ') 00 (x/2)V 2¢1 (0, 0; qV+ 1; q, -x2/4), q;q 00 (qv+ 1. q) ( x2qv+ J~2) (x; q) = (q; ;)0000 (x/2t 0¢1 - ; qV+ 1; q, - - 4 - , J~l) (x; q)

=

1)

J~3)(x;q)

=

(qv+ 1. q) ( \ 00 (x/2t 1¢1(0;qv+1;q,qx2/4), q;q 00

where 0 < q < 1. The above notations for the q-Bessel functions are due to Ismail [1981, 1982, 2003cl. Show that J~2)(x;q)

= (-x2/4;q)ooJ~1)(x;q),

Ixl < 2,

(Hahn [1949c])

and lim J~k)(x(l- q);q)

q--+1

= Jv(x),

k

= 1,2,3.

1.25 For the q-Bessel functions defined as in Exercise 1.24 prove that (i)

qV ik) (x·, q) = 2(1 -X qV) fk) (x· q) _ ik) (x· q) k = 1 , 2·, v+ 1 v' v-I"

(ii)

J~l) (xq!; q) = qv/2 (J~l) (x; q) + ~J~~ 1 (x; q)) ;

(iii)

J~l)(xq!;q)

(iv) qV+

1

=

J~~ 1 (xq1/2; q) =

q-v/2 (J~l)(x;q) - ~J~~l(X;q)). 2(1 - qV) J~3) (x; q) -

J~~l (x; q).

X

1.26 (i) Following Ismail [1982]' let

fv(x) = J~l) (x; q)J£12 (xq!; q) - J£12 (x; q)J~l) (xq!; q). Show that

and deduce that, for non-integral v,

fv(x)

=

q-v/2(qv, q1-v; q)oo/(q, q, _x 2/4; q)oo.

(ii) Show that

9v(qX)

+ (x2 /4 -

qV - q-v)9v (X)

+ 9v(xq-1) = 0

Exercises

31

with gl/(x) = J~3)(xql//2;q2) and deduce that gl/ (X) g-I/ ( xq

-1 )

- g-I/

() X

(

gl/ xq

-1) _

-

( 21/

q ,q

1-21/. 2)

,q (2 2. 2) q ,q ,q

00

q

1/( 1/-1)

.

00

(Ismail [2003c]) 1.27 Show that 00

n=

-CX)

00

Both of these are q-analogues of the generating function 00

L n=

t n In(x)

= ex (t-C

1

)/2.

-CX)

1.28 The continuous q-Hermite polynomials are defined in Askey and Ismail [1983] by H

(q; q)n i(n-2k)IJ ( I ) - ~ nxq-L..J( )( ) e , k= 0 q; q k q; q n-k

where x = cos B; see Szego [1926]' Carlitz [1955, 1957a, 1958, 1960] and Rogers [1894, 1917]. Derive the generating function n f:'~o Hn(xlq) (q; q)n t

1 =

(teiIJ, te-iIJ; q)oo'

It I < 1.

(Rogers [1894])

1.29 The continuous q-ultraspherical polynomials are defined in Askey and Ismail [1983] by

C ( . (31 ) - ~ ((3; q)k((3; q)n-k i(n-2k)IJ n x, q - L..J ( ) ( ) e , k= 0

q; q k q; q n-k

and 00

' " Cn(x; (3lq)t L..J n=O

n =

((3te iIJ , (3te- iIJ ; q)oo ( ·IJ -"IJ) te' te ' . q ' ,00

,It I < 1.

(Rogers [1895])

32

Basic hypergeometric series

1.30 Show that if

1.31 Let

~b

ml , ...

,mr are nonnegative integers, then

denote the q-difference operator defined for a fixed q by ~bJ(Z)

Then

~1

is the

~

= bJ(qz) - J(z).

operator defined in (1.3.20). Show that ~bXn

= (bqn - 1)xn

and, if

vn () z then

(al,"" ar; q)n (_1)(1+ s-r)n q(l+ s-r)n(n-l)/2 z n,

= (q,b 1 ,···,bs;q )n

(~~bdq~b2/q'" ~bs/q)Vn(z) = z(~al ~a2

... ~aJVn-l (zql+ s-r),

n

=

1,2, ...

Use this to show that the basic hypergeometric series

v(z)

= r¢s(al, ... , ar; b1 ,···, bs; q, z)

satisfies (in the sense of formal power series) the q-difference equation (~~bl/q~b2/q'" ~bs/q)V(z)

= Z(~al ... ~ar)v(zql+ s-r).

This is a q-analogue of the formal differential equation for generalized hypergeometric series given, e.g. in Henrici [1974, Theorem (1.5)] and Slater [1966, (2.1.2.1)]. Also see Jackson [191Od, (15)]. 1.32 The little q-Jacobi polynomials are defined by

Pn(x; a, b; q) = 2¢1 (q-n, abqn+ 1; aq; q, qx). Show that these polynomials satisfy the orthogonality relation

f

j= 0

~b~; ql j (aq)j Pn (qj; a, b; q)Pm( qj; a, b; q) q, q

J

0,

= { (q, bq; q)n(1- abq) (aqt (abl; q)= (aq,abq;q)n(l-abq2n+l) (aq;q)= '

if m =I- n, if m = n.

Exercises

33

(Andrews and Askey [1977 ]) 1.33 Show for the above little q-Jacobi polynomials that the formula n

Pn(x; c, d; q)

=

L ak,nPk(X; a, b; q) k= 0

holds with

_ _ k (k~l)(q-n,aq,cdqn+l;q)k ak,n - ( 1) q (q,cq,a bqk+I.) ,q k

3(Pl

[qk-n,Cdqn+k+l,aqk+l. ] cqk+1 , abq2k+2 ,q,q.

(Andrews and Askey [1977]) 1.34 (i) If m, ml, m2, ... , mr are arbitrary nonnegative integers and la- I qm+l-(m 1+···+m r)1 < 1, show that A..

r+ 2 'f'r+

I

[

a ' b,I b qml , ... , br qmr . a-I m+I-(ml+···+mr) ] bq 1+ m ,I, b ... , b ' q, q r

= (q, bq/a; q)oo(bq; q)m(bl /b; q)ml ... (br/b; q)mr bm1+ ... + mr-m (bq, q/a; q)oo(q; q)m(b l ; q)ml ... (b r ; q)mr X

q-m, b, bq/bl , ... ,bq/br ] [ r+ 2 ¢r+ I bq/ a, bqI -ml /b I,···, bqI-mr /b r ; q, q ;

(ii) if ml, m2, ... , mr are nonnegative integers and la-Iql-(m 1+ ... + mr ) 1< 1, Icql < 1, show that A..

r+ 2 'f'r+

=

I

[

a ' b,I b qml , ... , br qmr . a-II-(m1+···+mr) ] b b b ' q, q cq, I,···, r

(bq/a, cq; q)oo (b l /b; q)ml ... (br/b; q)mr bm1+ ... + mr (bcq, q/ a; q)oo (b l ; q)ml ... (b r ; q)mr X

c- I , b, bq/bl , ... ,bq/br ] r+ 2 ¢r+ I [ bq/ a, bqI -ml /b I , ... , bqI -m r /b r ; q, cq .

(Gasper [1981a]) 1.35 Use Ex. 1.2(v) to prove that if x and yare indeterminates such that xy = qyx, q commutes with x and y, and the associative law holds, then

(See Cigler [1979], Feinsilver [1982]' Koornwinder [1989], Potter[1950]' Schiitzenberger [1953], and Yang [1991]). 1.36 Verify that if x and yare indeterminates satisfying the conditions in Ex. 1.35, then

(i) (ii)

eq(y)eq(x) = eq(x + y), eq(x)eq(y) = eq(x + y - yx); Eq(x)Eq(y) = Eq(x + y), Eq(y)Eq(x) = Eq(x + y + yx).

34

Basic hypergeometric series (Fairlie and Wu [1997]; Koornwinder [1997], where q-exponentials with q- Heisenberg relations and other relations are also considered.)

1.37 Show that Eq(z;a)

(a 2;q2)= {

= (qa2.,q2) =

2 0, which reduces to Kummer's formula (1.8.2) by letting

---+ -00.

If the series (2.1.1) is well-poised and a2 = 1 + ~al' then it is called a very-well-poised series. Dougall's [1907] summation formulas F. [ 7 6

1

a, + ~a, b, c, d, e, -n . ~a, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a + n '

(1 + a)n(1 + a - b - c)n(1 + a - b - d)n(1 + a - c - d)n (1 + a - b)n(1 + a - c)n(1 + a - d)n(1 + a - b - c - d)n 38

1] (2.1.6)

2.1 Well-, nearly-, and very-well-poised series

39

when the series is 2-balanced (i.e, 1 + 2a + n = b + c + d + e), and F [

s

4

1

a, + ~a, b, c, d . ~a, 1 + a - b, 1 + a - c, 1 + a - d'

r(1 r(1

+ a - b)r(1 + a + a)r(1 + a - b -

+a +a -

c)r(1 c)r(1

1]

d)r(1 + a - b - c - d) b - d)r(1 + a - c - d)'

(2.1. 7)

when Re (1 + a - b - c - d) > 0, illustrate the importance of very-well-poised hypergeometric series. Note that Dixon's formula (2.1.5) follows from (2.1.7) by setting d = ~a. Analogous to the hypergeometric case, we shall call the basic hypergeometric series '" [a l ,a2, ... ,ar+ l . ] r+ 1 'f'r b1, ... , b ' q, Z r

(2.1.8)

well-poised if the parameters satisfy the relations qal = a2 bl = a3 b2 = ... = ar+lb r ; 1

(2.1.9)

1

very-well-poised if, in addition, a2 = qai, a3 = -qai; a nearly-poised series of the first kind if qal -=I=- a2 bl = a3 b2 = ... = ar+lb n

and a nearly-poised series of the second kind if qal = a2 bl = a3 b2 = ... = arb r - l

-=I=-

ar+lbr .

(2.1.10)

In this chapter we shall be primarily concerned with the summation and transformation formulas for very-well-poised basic hypergeometric series. To help simplify some of the displays involving very-well-poised r+lcPr series which arise in the proofs in this and the subsequent chapters we shall frequently replace

by the more compact notation (2.1.11) In the displays of the main formulas, however, we shall continue to use the r+l cPr notation, since in most applications one needs to know the denominator parameters. In all of the very-well-poised 6cPS, 8cP7 and lOcP9 series in Appendix II and Appendix III the parameters aI, a4, ... , ar+l and the argument z satisfy the very-well-poised balancing condition (a4as···ar+l)z= (±(alq)~r-3

(2.1.12)

with either the plus or minus sign, where (alq)~ is the principal value of the square root of alq. Thus, we will call a r+l Wr series very-well-poised-balanced (or, for brevity, VWP-balanced) if (2.1.12) holds. A special case of this condition was stated at the end of §2.4 in the first edition of this book. It follows from (2.1.12) that a VWP-balanced series is balanced (I-balanced) when z = q,

Summation, transformation, and expansion formulas

40

a balanced r+l Wr series is VWP-balanced when r is even, but it is not necessarily VWP-balanced when r is odd since then we only know that the squares of both sides of (2.1.12) (with z = q) are equal (see the comments in the first paragraph of §2.6). When r + 1 = 2k is an even integer, which is usually the case, (2.1.12) simplifies to the condition (a4a5'" a2k)z = (alq)k-2.

(2.1.13)

Notice that if we set a2k = ±(alq)!, then the 2kW2k-l series reduces to a series that satisfies (2.1.12) with r = 2k - 2 and the corresponding plus or minus sign. Since any well-poised r+l¢r series that satisfies the relations in (2.1.9) can be written as a very-well-poised series in the form

2k-l W 2k-2

1

1

1

1

r+5 W r+4(al; al, -al, qal, -qal, a2, a3,"" ar+l; q, z),

(2.1.14)

if follows that the very-well-poised balancing condition for the series in (2.1.14) is consistent with the well-poised balancing condition (ala2'" ar+dz

= -(±(alq)!y+l

(2.1.15)

for a well-poised r+l¢r series to be well-poised-balanced (WP-balanced). In particular, the 2¢1 series in the Bailey-Daum summation formula (1.8.1) is WP-balanced. Clearly, every VWP-balanced basic hypergeometric series is WP-balanced and, by the above observations, every WP-balanced basic hypergeometric series can be rewritten to be a VWP-balanced series of the form in (2.1.14). 2.2 A general expansion formula

Let a, b, c be arbitrary parameters and k be a nonnegative integer. Then, by the q-Saalschiitz formula (1. 7.2) 3¢2

(q-k, aqk, aq/bc; aq/b,aq/c; q,q) (C,ql-k/b;q)k

(b,c;q)k

(aq/b, cq-k fa; q)k

(aq/b, aq/c; q)k

so that ~

(b, c, q-n; qh

~ (q, aq/b, aq/c; q)k

k=O =

A k

Ln Lk

(/b k Ak aq c, aq k ,q -k.: q ) j (q.-n.) ,q k qj (b) ~ k=Oj=O (q, aq/b, aq/c, q)j(q, q)k aq

_ ~~ (aq/bc, aqi+j,q-i- j ; q)j (q-n;q)i+j . (bc)i+j -~~ ~ ~H j=Oi=O (q,aq/b,aq/c;q)j(q;q)i+j aq ~ (aq/bc, aqj, q-n; q) .

= ~

j=O

J

(q, aq/b, aq/c; q)j

.

(-I)J q -

(j) 2

(2.2.1)

2.3 A summation formula for a terminating n-j (qj-n, aq2j ;q)i

X" ~

(q,aqj;q)i

q

(bC)i+j

-i. J

-

aq

41

41>3

(2.2.2)

A+·

• J'

where {Ak} is an arbitrary sequence. This is equivalent to Bailey's [1949] lemma. Choosing (2.2.3)

X

r+ 2 1>r+ 1

[

aq 2j ,a1 qj ,a2qj , ... , arq j ,qj-n . ..

.

.,

b1qJ, b2qJ, ... , brqJ, br+ 1qJ

bcz ] q, ------:::j:l . aqJ

(2.2.4)

This is a q-analogue of Bailey's formula [1935, 4.3(1)]. The most important property of (2.2.4) is that it enables one to reduce the r+41>r+3 series to a sum of r+21>r+1 series. Consequently, if the above r+21>r+1 series is summable for some values of the parameters then (2.2.4) gives a transformation formula for the corresponding r+41>r+3 series in terms of a single series. 2.3 A summation formula for a terminating very-well-poised 41>3 series Setting b = qa!, c = -qa! and ak obtain from (2.2.4) that 1

41>3

[

=

bk , k

=

=

aqn+l, we

1

] a, qa 2" , -qa 2" , q-n ! ! n+1 ;q,z a 2 , -a 2 , aq

~ (-q-1,q-n;q)j(a;qhj(

= ~

1

1

j=O (q, a2", -a2", aqn+1; q)j

x

1,2, ... , r, br+ 1

21>1

). _(j) qz Jq 2

(aq2j,qj-n;aqHn+\q,_zq1-j).

(2.3.1)

If we set z = qn so that the above 41>3 series is VWP-balanced, then the 21>1 series on the right of (2.3.1) can be summed by means of the Bailey-Daum summation formula (1.8.1), which gives A,

(

2'1'1 aq

2j

,q

j-n. j+n+1. _ 1+n- j ) _ , aq , q, q -

( _q. q) (aq2j+1 aq2n+2. q2) '00 , '00 ( aqn +J·+1 ,-q1+ n _.) . J; q 00

(2.3.2) Hence, using the identities (1.2.32), (1.2.39) and (1.2.40), and simplifying, we obtain the transformation formula 1 1 n] a, qa2", -qa2" , q-n [ 41>3 ! ! n+1; q, q a 2 ,-a 2 ,aq (aq, -q; q)n

(qa !2 , -qa !.) 2 ,q n

A,

2'1'1

(-n -1 -n ) q , -q ; -q ; q, q .

(2.3.3)

42

Summation, transformation, and expansion formulas

Clearly, both sides of (2.3.3) are equal to 1 when n = O. By (1.5.3) the 2¢1 series on the right of (2.3.3) has the sum(ql-n; q)n (_q-l j (_q-n; q)n when n = 0,1, .... Since (ql-n; q)n = 0 unless n = 0, it follows that

r

1

4¢3

[

1

a, qa"2 , -qa"2 , q-n n] 1 1 n+l; q, q a 2 , -a 2 , aq

= 6n ,o ,

(2.3.4)

where m=n, m

(2.3.5)

i- n,

is the Kronecker delta function. This summation formula will be used in the next section to obtain the sum of a 6¢5 series. 2.4 A summation formula for a terminating very-well-poised 1

6¢5

1

series

1

Let us now set al = qa"2, a2 = -qa"2, b1 = a"2, b2 ak = bk , for k = 3,4, ... , r. Then (2.2.4) gives 1

rf..

[

6'i"5

=

=

1

-a"2, br+1 = aqn

+1

and

1

a, qa"2, -qa"2, b, c, q-n ] 1 1 +1 ; q, z a"2, -a"2, aqjb, aqjc, aqn

~ (aqjbc,qa!,-qa!,q-n;q)j(a;qhj (_bacqz)j q_(~) ~ j=O ( q,a 12 ,-a 12 ,aq jb ,aq j c,aq n+1.) ,q j X4¢3

[

aq2j,qJ+la!,_qj+la!,qj-n. bCZ] ·1 ·1 . 1 ,q'------:::t=1. qJ a"2 , _qJ a"2 , aqJ+n+ aqJ

If z = aqn+1 jbc, then we can sum the above obtain the summation formula rf..

6'i"5

[

4¢3

(2.4.1)

series by means of (2.3.4) to

a, qa!, -qa!, b, c, q-n . a qn +1 ] 1 1 +1 , q, - a"2, -a"2, aqjb, aqjc, aqn bc

(aqjbc,qa!,-qd,q-n;q)n (a;qhn (_1)nqn(n+l)/2 (q, a!, -a!, aqjb, aqjc, aqn+l; q)n (aq, aqjbc; q)n (aqjb, aqj c; q)n .

(2.4.2)

Note that the above 6¢5 series is VWP-balanced and that this summation formula reduces to (2.3.4) when bc = aq. In the next two sections, like climbing the steps of a ladder, we will use (2.4.2) to extend it to a transformation formula and a summation formula for very-well-poised S¢7 series. 2.5 Watson's transformation formula for a terminating very-well-poised

S¢7

series

We shall now use (2.4.2) to prove Watson's [1929a] transformation formula for a terminating very-well-poised S¢7 series as a multiple of a terminating

43

2.6 Jackson's sum balanced

4¢3

series:

89 series, which is both balanced and VWP-balanced, into a series of the same type: 1O¢>9

[

1

a, qa!, -qa!, b, c, d, e, j, Aaqn+1 /ej, q-n 1

1

a '5. , -a '5. ,aq/b, aq/c, aq/d, aq/e, aq/ j, ejq-n /A, aqn+

_ (aq, aq/ej, Aq/e, Aq/ j; q)n - (aq/ e, aq/ j, Aq/ ej, Aq; q)n

1O¢>9

;q,q

1

[A, qA!, -qA!, Ab/a, Ac/a, Ad/a, A! , -A! , aq/b, aq/ c, aq/ d,

e,j, Aaqn+1/e j,q-n, ] Aq/e, Aq/ j, ejq-n la, Aqn+1 ; q, q ,

(2.9.1)

where A = qa 2 /bcd. To derive this formula, first observe that by (2.6.2) 8¢>7

[

A, qA!, -qA!, Ab/a, Ac/a, Ad/a, aqm, q-m 1

1

A'5. ,-A'5., aq/b, aq/c, aq/d, Aq

1 -m

la, Aqm+

1 ;

1

q, q

Summation, transformation, and expansion formulas

48

(b, c, d, Aq; q)m (aq/b, aq/ c, aq/ d, a/ A; q)m '

(2.9.2)

and hence the left side of (2.9.1) equals ~ (a; q)m(l - aq2m)(e, I, Aaqn+l /el, q-n; q)m(a/ A; q)m m ~ m=O (q;q)m(l- a)(aq/e,aq/I,elq - n/A,aqn+l;q)m(Aq;q)m q

f

(A;q)j(l- Aq2j)(Ab/a,Ac/a,Ad/a,aqm,q-m;q)j j x j=O (q; q)j(l - A)(aq/b, aq/c, aq/d, Aql-m la, Aqm+l; q)j q

=

tf

(a; q)m+j(1- aq2m)(e, I, Aaqn+l /el, q-n; q)m m m=Oj=O (q;q)m-j(l- a)(aq/e,aq/I,elq-n/A,aqn+l;q)m q x (a/A; ~)m-j(A; :)j(l - Aq2j)(Ab/a, Ac/a, Ad(a; q)j (~)j (Aq, q)m+j(q, q)j(l - A)(aq/b, aq/c, aq/d, q)j A

=

t

~A; q)j(l - Aq2j)(Ab/a, Ac/a, Ad/a, e, I, Aaqn:~el, q~:l~)j . j=O (q,q)J(l- A)(aq/b,aq/c,aq/d,aq/e,aq/I,elq /A,aq ,q)J

aq)j (aq·qh· x (\ (A:) J /\ q, q 2J

8W7

( 2··· .1 . ) aq J;eqJ,lqJ,a/A,Aaqn+J+ /el,qJ-n;q,q , (2.9.3)

where the 8 W 7 series is defined as in §2.1. Summing the above 8 W 7 series by means of (2.6.2) and simplifying the coefficients, we obtain (2.9.1). It is sometimes helpful to rewrite (2.9.1) in a somewhat more symmetrical form: lOW9 (a; b, c, d, e, I, g, h; q, q)

(aq, aq/el,aq/eg, aq/eh, aq/Ig, aq/lh, aq/gh, aq/elgh; q)oo (aq/e,aq/I,aq/g,aq/h,aq/elg, aq/elh, aq/egh,aq/Igh; q)oo x lO W g (qa 2/bcd; aq/bc, aq/bd, aq/cd, e,J, g, h; q, q) , (2.9.4) where at least one of the parameters e, I, g, h is of form q-n, n and

=

0,1,2, ... , (2.9.5)

2.10 Limiting cases of Bailey's transformation formula

lO7 series by (2.6.2) we find that the sum over n in (2.12.8) equals lOW9(a;b,c,d,e,j,g,h;q,q) which is, of course, balanced by virtue of (2.12.4). Equating the expression in (2.12.2) with the sum of those in (2.12.5) and (2.12.8), and simplifying the coefficients, we finally obtain Bailey's [1947b] four-term transformation formula 1 1 A-. [ a,qa'2,-qa'2,b,c,d,e,j,g,h ] 10'1'9 a'2,-a'2,aq 1 1 / / / / / / / ; q, q b,aq c,aq d,aq e,aq j,aq g,aq h (aq,b/a,c,d,e,j,g,h,bq/c,bq/d;q)= 2 + (b q/a,a/b,aq/c, aq/d,aq/e,aq/j,aq/g, aq/h, bc/a, bd/a; q)= (bq/ e, bq/ j, bq/ g, bq/ h; q)= x~-,;---,--:-=::..,::--:-,~,-,-:-:-:--:-.:....::..:,.(be/a, bj la, bg/a, bh/a; q)= b2la, qba-~, -qba-~, b, bc/a, bd/a, be/a, bj la, bg/a, bh/a x 101>9 [ 1 1 ;q,q ba- '2, -ba- '2, bq/ a, bq/ c, bq/ d, bq/ e, bq/ j, bq/ g, bq/ h

1

(aq, b/a, Aq/j,Aq/g, Aq/h, bj/A,bg/A,bh/A; q)= (Aq,b/A, aq/j,aq/g, aq/h,bj/a, bg/a, bh/a;q)=

x

101>9

[

A,qA~,-qA~,b,Ac/a,Ad/a,Ae/a,j,g,h 1

1

;

A'2, -A'2, Aq/b, aq/c, aq/d, aq/e, Aq/ j, Aq/g, Aq/h

1

q, q

(aq,b/a,j,g,h,bq/j,bq/g,bq/h,Ac/a,Ad/a;q)=

+ (b 2q/A,A/b,aq/c,aq/d,aq/e,aq/j,aq/g,aq/h,bc/a,bd/a;q)= (Ae/a,abq/Ac,abq/Ad,abq/Ae;q)= x --'------'-:-::----:---',--::-.,-------:-----'-;-------:-:,----;-'------:------'--(be/a, bj la, bg/a, bh/a; q)=

x

101>9

[

b2 /A, qbA-~, -qbA-~, b, bc/a, bd/a, be/a, bj /A, bg/A, bh/A 1

1

;

bA-'2, -bA-'2, bq/A, abq/cA, abq/dA, abq/eA, bq/ j, bq/g, bq/h

1

q, q .

(2.12.9)

I

In terms of the q-integrals this can be written in a more compact form:

a

b

(qt/a, qt/b, ta-~, -ta-~, qt/c, qt/d, qt/e, qt/ j, qt/g, qt/h; q)= d 1

1

(t, bt/a, qta-'2, -qta-'2, ct/a, dt/a, et/a, jt/a, gt/a, ht/a; q)= a (b/a,aq/b, Ac/a, Ad/a, Ae/a, bj/A, bg/A, bh/A;q)= A (b/A, Aq/b, c, d, e, bj la, bg/a, bh/a; q)=

qt

Summation, transformation, and expansion formulas

58

1 b

x

A

1

1

(qt/ A, qt/b, tA -"2, -tA -"2, aqt/d, aqt/dA, aqt/e>.., qt/ f, qt/ g, qt/h; q)oo d 1 1 qt, (t, btl A, qtA -"2, -qtA -"2, ct/ a, dt/ a, et/ a, ft/ A, gt/ A, ht/ A; q)oo (2.12.10)

where A = qa 2/ede and a3q2 = bedefgh. Exercises 2.1 Show that

3 0; see Bailey [1935, 4.4(2)].

3.3 Three-term transformation formulas for

3¢2

series

In (2.10.10) let us replace a,b,e,d,e,j by AqN,BqN,C,D,E,FqN, respectively, and then let N ----; 00. In the resulting formula replace C, D, E, Aq/ B and Aq/ F by a, b, e, d and e, respectively, to obtain A.

3'1'2

[a, b, e. ~] _ (e/b, e/e; q)oo d , e ,q, a be - (e, e /b e; q) 00 (d/ a, b, e, de/be; q)oo

A.

3'1'2

[d/a, b, e. ] d, beq / e ,q, q

[e/b, e/ e, de/abe

]

(3.3.1) + (d, e, be/e, de/abe; q)oo 3¢2 de/be, eq/be ; q, q , Ide/abel < 1, and be/e is not an integer power of q. This expresses a 3¢2

where series of type II in terms of a 3¢2 series of type 1. As a special case of (3.3.1), let a = q-n with n = 0,1,2, .... Then A.

3'1'2

[q-n, b, e . de qn ] d, e ' q, ----,;;;-

_ (e/b, e/e; q)oo

- (e, e /b e; q)

00

(b, e; q)oo

+ (e, be/e; q)oo

A.

3'1'2

[b, e, dqn . ] / ' q, q beq e, d

(de/be; q)n ¢ [e/b, e/ e, deqn /be. ] (d; q)n 3 2 eq/be, de/be ,q, q .

(3.3.2)

Setting n = 0 in (3.3.2) gives the summation formula (2.10.13). We shall now obtain a transformation formula involving three 3¢2 series of type II. We start by replacing a, b, e, d, e, j in (2.11.1) by AqN, BqN, C, D, E, FqN, respectively, and then taking the limit N ----; 00. In the resulting formula we replace C, D, E, Aq/ B, Aq/ F by a, b, e, d, e, respectively, and obtain 3¢2

b, e de] [ a,d,e ;q, abe

_ (e/b, e/ e, eq/ a, q/ d; q)oo ¢ [e, d/ a, eq/ e . bq ] - (e, eq/d, q/a, e/be; q)oo 3 2 eq/a, beq/e' q, d (q/ d, eq/ d, b, e, d/ a, de/beq, beq2 / de; q)oo ¢ [a q/ d, bq/ d, eq/ d . de ] - (d/q, e, bq/d, eq/d, q/a, e/be, beq/e; q)oo 3 2 q2/d, eq/d ,q, abe ' (3.3.3) provided Ibq/dl < 1, Ide/abel < 1 and none of the denominator parameters on either side produces a zero factor. If Iql < Ide/abel < 1, then A.

3'1'2

[a,d,eb, e .,q, abe de ]

_ (e/b, e/e, q/d, bq/a, eq/a, abeq/de; q)oo ¢ [q/a, d/a, e/a. abeq ] (e, e/be, q/a, bq/d, eq/d, beq/e; q)oo 3 2 bq/a, eq/a ,q, de

74

Additional summation, transformation, and expansion formulas

-

(b, c, ql d, dl a, eql d, delbcq, bcq2 I de; q)oo rP [aql d, bql d, cql d . de ] (e, elbc, qla, bqld, cqld, bcqle, dlq; q)oo 3 2 q2ld, eqld ,q, abc ' (3.3.4)

by observing that from (3.2.7)

C,dla,cqle. bq] 3rP2 [ cq I a, bcq I e ,q'-d _ (abcql de, bql a; q)oo rP [ql a, dl a, el a . abcq ] - (bcqle,bqld;q)oo 3 2 bqla,cqla ,q, de . If we set e = '\c in (3.3.3), let c ----+ 0 and then replace d and ,\ by c and abzlc, respectively, where Izl < 1, Ibqlcl < 1, then we obtain

2rPl (a, b; c; q, z )

=

(abzlc, qlc; q)oo ( I I ) 2rPl (CI a, cq I abz; cq I az; q, bqI c ) az c, q a; q 00

(b, qlc, cia, azlq, q2 laz; q)oo

(cI q, bqI c, qI a, az I c, cq I az; q)

,/, (aq I c, bqI c; q21 c; q, z. )

2'1'1

(3.3.5 )

00

Sears' [1951c, p. 173] four-term transformation formulas involving 3rP2 series of types I and II can also be derived by a combination of the formulas obtained in this and the previous section. Some of these transformation formulas also arise as special cases of the more general formulas that we shall obtain in the next chapter by using contour integrals.

3.4 Transformation formulas for well-poised 3rP2 and very-well-poised 5rP4 series with arbitrary arguments Gasper and Rahman [1986] found the following formula connecting a wellpoised 3rP2 series with two balanced 5rP4 series:

a, b, c aqx] 3rP2 [ aq I b, aq I c ;q, -bc _ (ax; q)oo ( ) x; q 00

-

,/, [a!, -a!, (aq)!, -(aq)!, aqlbc. ] ,q, q aqlb, aqlc, ax, qlx

5'1'4

(a,aqlbc,aqxlb,aqxlc;q)oo

+ (aqlb, aqlc, aqxlbc, X-I; q)oo x

,/, [xa!,-xa!,x(aq)!,-x(aq)!,aqxlbc ] ; q, q . aqxlb,aqxlc,xq,ax 2

5'1'4

(3.4.1)

Convergence of the 3rP2 series on the left requires that laqxlbcl < 1. It is also essential to assume that x does not equal q±j, j = 0,1,2, ... , because of the factors (x; q)oo and (x-\ q)oo appearing in the denominators on the right side of (3.4.1). Note that if either a or aqlbc is 1 or a negative integer power of q,

3.4 Transformation formulas of well-poised

3¢>2

and

5¢>4

series

75

then the coefficient ofthe second 5¢>4 series on the right vanishes, so that (3.4.1) reduces to the Sears-Carlitz formula (Ex. 2.26). An important application of (3.4.1) is given in §8.8. To prove (3.4.1) we replace d by dqn in (2.8.3) and then let n ----7 00. This gives ¢>

[

3 2

a, b, e . d] _ (bed/aq; q)oo aq/b, aq/e' q, ~ - (bed/qa 2; q)oo

x lim 12W11 (a 2q1 - n /bed; a!, -a!, (aq)!, -(aq)!,aql-n/be, n---+oo aql-n /bd, aql-n /ed, a 3q3-n /b 2e 2d 2 , q-n; q, q) . (3.4.2) To take the limit on the right side of (3.4.2) it suffices to proceed as in (2.10.9) to obtain

'" [d,-d,(aq)!,-(aq)!,aq/be ] ;q,q aq/b,aq/e,bed/aq,a 2q2/bed

=

5'1'4

bed (bed/ a 2 , bd/ a, ed/ a, aq/be, a; q)oo qa 2 (d/a,aq/b,aq/e,bed/a,a 2q2/bed;q)oo

x

5¢>4

[

d/a, bed/qa~, -bed/qa~ , bed/q! a~, -bed/q! a~ bd/a,ed/a,bed/a2,b2e2d2/qa2

]

;

q, q . (3.4.3)

Using this in (3.4.2) and replacing d by qxa 2/be, we get (3.4.1). If we now replace d by dqn in (2.8.5) and then let n ----7 00, we obtain the transformation formula '" [ a,qa!,-qa!,b,e. x(a q )!] 1 1 ,q, - b a"2, -a"2, aq/b, aq/e e

5'1'4

(1 - x 2 )(xq(aq)!; q)oo (x(aq)-!;q)oo

x

5¢>4

[

(aq)!, -(aq)!, qa!, -qd, aq/be 1

1

aq/b,aq/e,xq(aq)"2,q(aq)2/x

] ;q,q

(aq,aq/be,x(aq)!/b,x(aq)!/e;q)oo

+ (aq/b, aq/e, x(aq) "2 /be, (aq)2 Ix; q)oo 1

x

X,

5¢>4 [

1

-x, xq!, -xq! , x(aq)! /be 1

1

1;

x(aq)"2/b,x(aq)"2/e,x(q/a)"2,qx 2

] q, q .

In terms of q-integrals formulas (3.4.1) and (3.4.4) are equivalent to 3¢>2

[

a,b,e aqx] aq/b, aq/e; q, bc

(3.4.4)

76

Additional summation, transformation, and expansion formulas (a, aq/bc; q)oo s(l - q) (q, aq/b, aq/ c, q/ x, X; q)oo

x

1

(qu/xs, qui s, aqu/bs, aqu/cs, axu/ s; q)oo

8

8X

1

1

1

1

(ua"2 / s, -ua"2 / s, u(aq) "2 / s, -u(aq) "2 / s, aqu/bcs; q)oo

d qU,

(3.4.5) and 1 -qa"2, 1 b, c. a, qa"2,

[

5¢4

x(aq) "21

]

q, - b a"2, -a"2, aq/b, aq/c c 1

1

,

(1 - x 2) (aq, aq/bc; q)oo s(l - q)(q, aq/b, aq/c, x(aq)-~, q(aq)~ Ix; q)oo

x

1

(uq(aq) ~ / sx, qui s, aqu/bs, aqu/cs, uxq(aq) ~ / S; q)oo d

8

1

8x(aq)-~ (u(aq) "2 /

1

1

qU,

1

s, -u(aq) "2 / s, uqa"2 / s, -uqa"2 / s, aqu/bsc; q)oo

(3.4.6) respectively, where s -=I=- 0 is an arbitrary parameter. If we now set c = (aq)~ in (3.4.5), replace x by x/b(aq)~, and use (2.10.19), then we get 2

2¢1(a, b;aq/b; q,qx/b )

x

8¢7

[

(xq/b, aqx 2/b 2; q)oo

= (aqx /b ,qx2/F) ,q

00

ax/b, q(ax/b)~, -q(ax/b)~, x, a!, -a~, (aq)~, -(aq)~

qx] q, 2 (ax/b) "2 ,-(ax/b)"2, aq/b, xqa"2 /b, -xqa"2 /b, x(aq) "2 /b, -x(aq) "2 /b b 1

1

1

1

1

1;

(3.4.7) provided Iqx/b 2 < 1 when the two series do not terminate. Similarly, setting c = (aq)~ and replacing x by x/bq in (3.4.6) we obtain 1

¢ [a,qa~,-qa!,b . ....::...] _ (ax 2/b 2 ,x/qb;q)00 a~, -a~, aq/b ,q, qb 2 - (aqx/b, x 2/qb 2; q)oo

4 3

X 8¢7

[

ax/b, q(ax/b)~, -q(ax/b)~, (aq)~, -(aq)~, qa~, -qa~, x 1

1

1

1

1

1

(ax/b) "2 ,-(ax/b)"2, x(aq) "2 /b, -x(aq)"2 /b, xa"2 /b, -xa"2 /b, aq/b

x ] ;

q,

-2

qb

(3.4.8) provided Ix/qb 2

1

< 1 when the series do not terminate.

'

3.5 Quadratic transformations

77

3.5 Transformations of series with base q2 to series with base q If in Sears' summation formula (2.10.12) we set b = -c, e = -q, replace a by aqr, r = 0, 1, 2, ... , multiply both sides by (x 2 ,y2;q2)r b2r qr 2 2 2 (-q; q)r(x y b ; q2)r

and then sum over r from 0 to (-1,-q,ab,-ab,b 2 ;q)00 ( a, b, -a, - b, b, - b ; q) 00 _

-

we get

00,

A,.

a 2,x 2,y2 . 2 b2] 2b2 2 2b2 , q , q a, x y

[

3'1-'2

b -b· q) J" j (_q ' ab2., q) 00 00 (a " , ( b -b·) b2.) "q a" ,qooj=o q, q,a ,qJ

L (_

+ (-q,-ab2;q)00 ·) ( -a, - b, b, q

3¢2

~ (-a,-b,b;q)j

j x2 , y [ -q 2, 2 . 2 2] x 2y 2b2 q2-2 j jb 2 , q , q ' j

A,.

[

~ ( q, -q, -a b2., q ) J" q 3'1-'2 J=O

00"

2 2]

q-2 j ,x2,y2 . x 22b2 y , q2-2 j jb2 ,q,q

(3.5.1) assuming that Iqb 2 < 1 when the series on the left is nonterminating. Since the two 3¢2 series on the right side can be summed by the q-Saalschiitz formula (1.7.2) with the base q replaced by q2, it follows from (3.5.1) that 1

_ (-a, ab 2 ; q)oo (-1, b2 ; q)oo

-

+

A,.

5'1-'4

(a, -ab 2; q)oo (-1,b2;q)00

[a, bx, -bx, by, -by. ] -q, ab2 , bxy, -bxy' q, q

A,.

5'1-'4

[-a, -bx, bx, -by, by. ] -q,-ab2,-bxy,bxy,q,q .

(3.5.2)

Note that one of the terms on the right side of (3.5.2) drops out when a ±q-n, n = 0,1,2, .... Setting y = ab and using (2.10.10) gives 2¢1

( 2

2

2 2 4

2

2)

a , x ; a x b ; q , qb

X 8 W 7 (ab 2x 2 jq;

=

(b2,a2b2x2;q2)00(ab2,b2x2;q)00

= (a 2b 2, b2x,q 2. 2) (b 2 b2 2. ) 00 ,a x,q 00

a, x, -x, bx, -bx; q, ab 2),

(3.5.3)

where Iqb 2 < 1 and lab2 < 1 when the series do not terminate. Byapplying Heine's transformation formula (1.4.1) twice to the 2¢1 series above and replacing b by q! jb we find that 1

2¢1 (a 2 , b2 ; a 2q2 jb 2 ; q2,

1

x 2q2 jb4 )

(qa2x2jb2,q2a2x2jb4;q2)00(aqjb2,qx2jb2;q)00 (qx2jb2,q2x2jb4;q2)00(qa2jb2,aqx2jb2;q)00

78

Additional summation, transformation, and expansion formulas

(3.5.4) provided laq/b 2 < 1 and Ixq/b 2 < 1 when the series do not terminate. This formula was derived by Gasper and Rahman [1986], and a terminating version of it was given earlier by Verma [1980]. Application of the transformation formula (2.10.1) to the 8(P7 series on the right of (3.5.4) yields an equivalent formula 1

1

(qa 2x 2/b 2 , q2a 2x 2/b 4 ; q2)oo (-xq~ /b, -axq~ /b 3 ; q)oo (qx 2/b 2 , q 2x 2/b 4 ; q2)oo( -axq~ /b, -a2xq~ /b 3 ; q)oo x

8eP7 [

-xa2q~ /b 3 , q( -xa2q~ /b 3 ) ~ , 2

1

3

1

(-xa q2/b )2,

aq~ /b, -aqx/b 2 ,

1 -aq2/b, aqx/b 2 ,

2 -aq/b,

axq~/b,

-q( -xa2q~ /b 3 ) ~, a, 2

1

3

III

1

3

3

-(-xa q2/b )2, -axq2/b,

-xq2 /b . xq2 2 2 qa /b ,q'--b- ,

(3.5.5)

where Ixq/b 2 < 1 and Ixq~ /bl < 1 when the series do not terminate. It is clear that formula (3.5.5) is a q-analogue of the quadratic transformation formula 1

2Fl (a, b; 1 + a - b; x 2)

=

(1 - x) -2a 2Fl (a, a

1 2

+- -

b; 2a

+1-

2b; (

-4x I-x

)2).

(3.5.6) We shall now prove the following transformation formula due to Jain and Verma [1982]:

eP [ 10 9

a~,

a, q2a~, -q2a~, b, e, eq, d, dq, e, eq 2 a 3q3 ] -a~, aq2/b, aq2/e, aq/e, aq2/d, aq/d, aq2/e, aq/e; q , be2d 2e 2

= (aq/e, aq/d, aq/e, aq/ede; q)oo

5eP4

(

aq

)12, - (aq )12, aq /b ,ed e / a ; q, q

+

(aq2, a 3q3/e 2d 2e 2; q2)oo(e, d, e, a 2q2 /bede; q)oo (aq2/b,a3q3/be2d2e2;q2)oo(aq/e,aq/d,aq/e,ede/aq;q)oo

x

5eP4

[

1

[ (aq/b) ~, -(aq/b) ~,e, d, e

(aq, aq/ed, aq/ee, aq/de; q)oo

1

(a3q3/be2d2e2)~,-(a3q3/be2d2e2)~,aq/ed,aq/ee,aq/de ; q,

q

,

(a3q3/e2d2e2)~,-(a3q3/e2d2e2)~,a2q2/bede,aq2/ede

(3.5.7)

3.5 Quadratic transformations

79

with the usual understanding that if the 104>9 series on the left does not terminate then the convergence condition la3q3/bc2d2e21 < 1 must be assumed to hold. First we rewrite (2.10.12) in the form

(aq4n+l, aq/cd, aq/ce, aq/de; q)oo (cq2n, dq2n, eq2n, aql-2n / cde; q)oo

+ (a 2q2n+2/cde; q)oo (cdeq2n-l fa; q)oo

00

~

00

(cq2n, dq2n, eq2n; q)r

r

~ (q, aq4n+l, cdeq2n / a; q)r q

(aq/cd, aq/ce, aq/de; q)r qr 2 (q, a q2n+2 /cde, aq2-2n /cde; q)r

(c, d, e; qhn (aq/c, aq/d, aq/e; q)oo (c, d, e; q)oo (aq/c, aq/d, aq/e; qhn'

(3.5.8)

where n is a nonnegative integer. Using (1.2.39) and (1.2.40), multiplying both sides of (3.5.8) by

(a,b;q2)n(1-aq4n) ( a3q3 )n (q2, aq2/b;q2)n(1-a) bc2d 2e2 ' and summing over n from 0 to

00,

we get

loVVg(a;b,c,cq,d,dq,e,eq;q2,a3q3/bc2d2e2)

f

(aq, aq/cd, aq/ce, aq/de; q)oo (a, b; q2)n(1- aq4n) - (aq/c, aq/d, aq/e, aq/cde; q)oo n=O (q2, aq2/b; q2)n(1- a) (c, d ,e; q) 2n n(2n-l) ( aq 3 ) n x (cde/a; qhn(aq; q)4n q b

+

34>2

[ cq2n, dq2n, eq2n ] aq4n+1, cdeq2n /a; q, q

(c,d, e,a 2q2/cde; q)oo ~ (a,b;q2)n(1 _aq4n) (aq/ c, aq/ d, aq/ e, cde/ aq; q)oo (q2, aq2/b; q2)n (1- a)

t:o

(cde/aq; qhn ( a3q3 ) n x (a 2q2/cde;q)2n bc2d2e2

34>2

[aq/cd, aq/ce, aq/de ] a2q2n+2/cde,aq2-2n/cde;q,q . (3.5.9)

The first double series on the right side of (3.5.9) easily transforms to

f

(c,d,e;q)m qm 6 VV5 (a;b,ql-m,q-m;q2,aq2m+l/b) , m=O (q, aq, cde/a; q)m which, by (2.4.2), equals 54>4

[

c, d, e, (aq/b)!, -(aq/b)! 1

1

aq/b, cde/a, (aq)2, -(aq)2

;

1

q, q .

Similarly we can express the second double series on the right side of (3.5.9) as a single balanced 54>4 series. Combining the two we get (3.5.7). The special case of (3.5.7) that results from setting e = (aq)! is particularly interesting because both 54>4 series on the right side become balanced 44>3

80

Additional summation, transformation, and expansion formulas

series which, via (2.10.10), combine into a single we have the formula 8cP7

[

8cP7

series with base q. Thus

a, q2a~, -q2a~, b, e, eq, d, dq

2 a 2q2 ] q , b 2d2 a>, -a>, aq /b, aq /e, aq/e, aq /d, aq/d e 1

1

2

2

2

;

(aq, aq/be, aq/ed, -aq/ed, aq/db~, -aq/db~; q)oo (aq/b, aq/ e, aq/d, -aq/d, aq/edb~, -aq/edb~; q)oo

x

8cP7

[

-aid, q( -a/d)~, -q( -a/d)~, e, b~, -b~, (aq)~ /d, -(aq)~ /d

aq ] q, -b ' (-a/d) >, -( -a/d» , -aq/ed, -aq/db> , aq/db> , -(aq) > , (aq) > e 1

1

1

1

1

1

;

(3.5.10) where la 2q2/be2d 2

1

< 1 and

laq/bel

< 1 when the series do not terminate.

3.6 Bibasic summation formulas Our main objective in this section is to derive summation formulas containing two independent bases. Let us start by observing that when d = a/be Jackson's 8cP7 summation formula (2.6.2) reduces to the following sum of a truncated series

t

k=O

1 - aq2k (a, b, e, a/bc; q).k 1 - a (q, aq/b, aq/e, beq, qh

l

=

(aq, bq, eq, aq/be;?)n , (q, aq/b, aq/e, beq, q)n

where n = 0,1, .... Notice that this series telescopes, for if we set and (aq,bq,eq,aq/be;q)k O"k = (q,aq/b,aq/e,beq;q)k for k = 0,1, ... , and apply the difference operator Uk-l to O"k, then we get A

1...l.00k

=

~

defined by

(3.6.1) 0"-1

~Uk

(1 - aq2k)(a, b, c, a/bc; qh k q (1 - a)(q, aq/b, aq/e, beq; qh '

= 0

(3.6.2)

= Uk (3.6.3)

which gives (3.6.1), since n

L~Uk = Un - U-l

(3.6.4)

k=O

for any sequence {Uk}. These observations and the bibasic extension Tk

=

(ap,bp;p)k(eq,aq/be;q)k (q,aq/b; q)k(ap/c, bep;p)k

(3.6.5)

of O"k were used in Gasper [1989a] to show that A I...l.~=

(1 - apkqk)(1 - bpkq-k) (a, b;p)k(e, a/bc; q)k k q (1 - a)(1 - b) (q, aq/b; qh(ap/c, bep; ph '

(3.6.6)

81

3.6 Bibasic summation formulas which, by (3.6.4), gave the indefinite bibasic summation formula

t

k=O

(1 - apkqk)(l - bpkq-k) (a, b;ph(e, a/be; qh qk (1 - a)(l - b) (q, aq/b; q)k(ap/e, bep;p)k

(ap, bP;P)n(eq, aq/be; q)n (q,aq/b;q)n(ap/c,bep;P)n

(3.6.7)

for n = 0,1, .... Notice that the part of the series on the left side of (3.6.7) containing the q-shifted factorials is split-poised in the sense that aq = b(aq/b) and e(ap/e) = (a/be) (bep) = ap, while the expression on the right side is balanced and well-poised since

(ap) (bp) (eq) (aq/be)

q( aq/b) (ap/ e) (bep)

=

and The b ----;

(ap)q

°

(bp)(aq/b)

=

=

(eq)(ap/e)

=

(aq/be)(bep).

case of (3.6.7)

..[!--. 1 - apkqk (a;p)de; q)k -k - - --:----,-----:---,,-:-----,-----:-----,----:--e = ~

k=O

1- a

(q;q)k(ap/e;p)k

(ap;P)n(eq; q)n -n e (q;q)n(ap/e;P)n

(3.6.8)

is due to Gosper. To derive a useful extension of (3.6.7), Gasper and Rahman [1990] set

(ap,bp;p)k(eq, ad2q/be; q)k Sk = (dq, adq /b;) q k (adp / e, bep / d;p )k for k

= 0, ±1, ±2, ... , and /}.Sk

=

(3.6.9)

observed that

Sk - Sk-l (ap, bp;ph-l (eq, ad 2q/be; qh-l (dq,adq/b; q)k(adp/e, bep/d;p)k x {(I - apk)(l - bpk)(l - eqk)(l - ad 2qk /be)

-(1 - dqk)(l - adqk /b)(l - adpk /e)(l - bepk /d)} d(l - e/d)(l - ad/be)(l - adpkqk)(l - bpk /dqk) (1 - a)(l - b)(l - e)(l - ad 2/be) (a, b;p)k(e,ad2/be; q)kqk x (dq, adq /b;) q k (adp / e, bep /d;) p k

(3.6.10)

Since (3.6.4) extends to n

L

/}.Uk = Un - U-'m-l,

(3.6.11)

k=-'m where we employed the standard convention of defining

a'm n

L k='m

ak =

{

+ a'm+l + ... + an,

0,

-(an+l

+ an+2 + ... + a'm-d,

m:::::n, m

= n+ 1,

m

~

n+2, (3.6.12)

82

Additional summation, transformation, and expansion formulas

for n, m = 0, ±1, ±2, ... , it follows from (3.6.10) that (3.6.7) extends to the indefinite bibasic summation formula ~ (l-adpkqk)(l-bpk/dqk)

~

(1 - ad)(l - bid)

k=-rn

(a,b;ph(e,ad 2 /be;q)k k (dq, adq/b; q)dadp/e, bep/d;p)k q

(1 - a)(l - b)(l - e)(l - ad 2 /be) d(l- ad)(l - b/d)(l - e/d)(l - ad/be) 2 (e/ad, d/be;p)rn+l(l/d, b/ad; q)rn+l} { (ap, bp;p)n(eq, ad q/be; q)n x (dq, adq/b; q)n(adp/e, bep/d; P)n - (l/e, be/ad2 ; q)rn+l (l/a, l/b; P)rn+l (3.6.13) for n, m = 0, ±1, ±2, ... , by applying the identity (1.2.28). Observe that (3.6.7) is the case d = 1 of (3.6.13) and that the right side of (3.6.9) is balanced and well-poised since

(ap) (bp) (eq)( ad 2 q/be) = (dq) (adq/b) (adp/ c) (bep/ d) and

(ap)(dq)

=

(bp)(adq/b)

=

(eq)(adp/e)

=

(ad 2 q/be)(bep/d).

It is these observations and the factorization that occurred in (3.6.10) which motivated the choice of Sk in (3.6.9). If Ipl < 1 and Iql < 1, then by letting n or m tend to infinity in (3.6.13) we find that (3.6.13) also holds with n or m replaced by 00. In particular, this yields the following evaluation of a bilateral bibasic series ~

~ k=-oo

(1 - adpkqk) (1 - bpk / dqk) (a, b; ph (e, ad2 /be; q)k k (1 - ad)(l - bid) (dq, adq/b; q)k(adp/e, bep/d;ph q

(1 - a)(l - b)(l - e)(l - ad 2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) 2 (e/ad, d/be;p)oo(1/d, b/ad; q)oo} { (ap, bp; p)oo (eq, ad q/be; q)oo x (dq, adq/b; q)oo(adp/e, bep/d; p)oo - (l/e, be/ad2 ; q)oo(1/a, l/b; p)oo ' (3.6.14) where Ipi < 1 and Iql < 1. In §3.8 we shall use the m =

°

case of (3.6.13) in the form

~ (1 - adpkqk)(l- bpk /dqk)

~

k=O

(1- ad)(l- bid)

(a, b;p)k(e, ad 2 /be; q)k k (dq, adq/b; qh(adp/e, bep/d;ph q

(1 - a)(l - b)(l - e)(l - ad2 /be) d(l - ad)(l - b/d)(l - e/d)(l - ad/be) (ap,bp;p)n(eq,ad 2 q/be;q)n X~'-----::---;-:,------'--,-----;-----::----,------'::-----;-:'------:-(dq, adq/b; q)n(adp/e, bep/d; P)n (1 - d)(l - ad/b)(l - ad/c) (1 - be/d) d(l - ad)(l - b/d)(l - e/d)(l - ad/be)·

(3.6.15)

There is no loss in generality since, by setting k = j - m in (3.6.13), it is seen that (3.6.13) is equivalent to (3.6.15) with n, a, b, e, d replaced by n +

3.6 Bibasic summation formulas

83

m, ap-m , bp-m, cq-m, dq-m, respectively. We shall also use the special case c = q-n of (3.6.15) in the form

t k=O

(1 - adpkqk)(l - bpk /dqk) (a, b;p)k(q-n, ad 2qn /b; qh k (1 - ad)(l - bid) (dq, adq/b; qh(adpqn, bp/dqn;ph q

(1 - d)(l - ad/b)(l - adqn)(l - dqn /b) (1 - ad)(l - d/b)(l - dqn)(l - adqn /b)' where n = 0,1, .... The d ----+ 1 limit case of (3.6.16) ~ (1 - apkqk)(l - bpkq-k)

~

(1- a)(l- b)

k=O

(3.6.16)

(a, b;ph(q-n, aqn /b; qh k _ 6 (3.6.17) (q, aq/b; q)dapqn, bpq-n;p)k q - n,O,

where 6n ,m is the Kronecker delta function and n = 0,1, ... , was derived independently by Bressoud [1988], Gasper [1989a], and Krattenthaler [1996]. If we replace n, a, band k in (3.6.17) by n - m, apmqm, bpmq-m and j - m, respectively, we obtain the orthogonality relation n

L

anjbjm

= 6n,m

(3.6.18)

j=m

with (-l)n+j(l- apJqj)(l- bpJq-j)(apqn,bpq-n;P)n_l anj = (q; q)n_j(apqn, bpq-n;p)j(bql-2n/a; q)n-j ,

(3.6.19) (3.6.20) This shows that the triangular matrix A = (anj) is inverse to the triangular matrix B = (b jm ). Since inverse matrices commute, by computing the jkth term of BA, we obtain the orthogonality relation

t

. k

(1 - apkqk)(l - bpkq-k) (apk+lqk+n, bpk+lq-k-n;p)j_k_l (q; q)n(q; q)j_k_n(aq2k+n /b; q)j-k-l

n=O

x

(1 - ~q2k+2n) (_ltqn(j-k-l)+e-~-n) = 6j,k,

(3.6.21)

which, by replacing j, n, a, b by n + k, k, ap-k-lq-k, bp-k-lqk, respectively, yields the bibasic summation formula

(1- ~) (1 - ~) t p

p

(aqk, bq-k;P)n_l(l - aq2k /b) k=O (q; q)k(q; q)n_k(aqk /b; q)n+l

(_l)kq(~)

=

6n 0 '

(3.6.22) for n = 0,1,... . The b ----+ limit case of (3.6.22) was derived in AI-Salam and Verma [1984] by using the fact that the nth q-difference of a polynomial in q of degree less than n is equal to zero. For applications to q-analogues of Lagrange inversion, see Gessel and Stanton [1983, 1986] and Gasper [1989a]. Formulas (3.6.17) and (3.6.22) will be used in §3.7 to derive some useful general expansion formulas.

°

84

Additional summation, transformation, and expansion formulas 3.7 Bibasic expansion formulas

One of the most important general expansion formulas for hypergeometric series is the Fields and Wimp [1961] expansion

(3.7.1) where we employed the contracted notation of representing al, ... , ar by aR , (al)n'" (ar)n by (aR)n, and n + al,"" n + ar by n + aR. In (3.7.1), as elsewhere, either the parameters and variables are assumed to be such that the (multiple) series converge absolutely or the series are considered to be formal power series in the variables x and w. Special cases of (3.7.1) were employed, e.g., in Gasper [1975a] to prove the nonnegativity of certain sums (kernels) of Jacobi polynomials and to give additional proofs of the Askey and Gasper [1976] inequalities that de Branges [1985] used at the last step in his proof of the Bieberbach conjecture. Verma [1972] showed that (3.7.1) is a special case of the expansion

oo oo (a)n+d,8)n+k B _ L (_x)n L L_ A n B n (xw)n n+k n! _ n!b+n)n k!b+2n+1h OO

n-O

n-O X

k

k=O

~ (-n)j(n+'Y)jA.w j ~ j=O

X

j!

() () a j j

,8

J

( ) 3.7.2

and derived the q-analogue

fAnBn (xw)n n=O (q; q)n

_Loo

-

L

(-x)n C2) oo (a,,8; q)n+k B k q n+k X n=O (q,'Yqn;q)n k=O (q,'Yq2n+l;qh X

t

(q-n,'Yqn;q)j Aj(wq)j. j=O (q,a,,8;q)j

(3.7.3)

To derive a bibasic extension of (3.7.3) we first observe that, by (3.6.17),

(3.7.4)

3.7 Bibasic expansion formulas

85

for m = 0,1, .... Hence, if Cr,m are complex numbers such that Cr,o = 1 for r = 0,1, ... , then B

r rX

00

=~ X

1_"W- 1q2r+2m ("(a- 1q2r;q)m("(pqr,apq-r;P)r 1 _ _W- 1q2r (q; q)m("(pqr+m, apq-r-m;P)r

q -mrBr+m Cr,mx r+ms:Um,O

00

00

= {;~

(1 _1'pnqn)(l _ apnq-n) (1 _1'a-1q2n+2k) (q; q)k(q; q)n("(pqn+k, apq-n-k;P)n

x ("(a-1qn+r+1; q)n+k-r-1 ("(pqr, apq-r;P)n_1 (q-n; q)r

x (_l)n B n+k C r,n+k-r xn+kqn(1+r-n-k l+G) (3.7.5) by setting j = n - rand m = n + k - r. Then by multiplying both sides of (3.7.5) by Arw r /(q; q)r and summing from r = to 00 we obtain Gasper's [1989a] bibasic expansion formula

°

f

n=O

AnBn (xw)n (q; q)n

=

f

n=O

(l-1'pnqn)(l - apnq-n) (-xtqn+G) (q; q)n

x'"' 00

1 - 1'a- 1q2n+2k

k

~ (q; q)k("(pqn+k, apq-n-k;P)n

X

Bn+kx

Ln

(-n q ; q) j (1'a -1 qn+j+1 ; q )n+k-j-1 j=O (q; q)j

x (1'pqj, apq-j; P)n-1AjCj,n+k-jw j qn(j-n-k),

(3.7.6)

where Cj,o = 1, for j = 0,1, .... Note that if p = q and Cj,m == 1, then (3.7.6) reduces to an expansion which is equivalent to

f

fAnBn (xw)n = (a, 1' qn+1/ a , a, (3; q)n n=O (q;q)n n=O (q,1' qn;q)n

(::.r a

("(q2n/a,qn+l"h/a,-qn+1yh/a,1/a,aqn,(3qn;q)kB k xL n+k X k=O (q, qn V1' /a, -q n V1' /a, 1'q2n+1; q)k 00

n

(-n n) q , 1'q ; q j A- (wq)j . ~ (q,1'qn+1/a ,q1-n/a ,a,(3;q)j J

X '"'

(3.7.7)

Verma's expansion (3.7.3) is the a ----7 00 limit case of (3.7.7). For basic hypergeometric series, (3.7.7) gives the following q-extension of (3.7.1)

86

Additional summation, transformation, and expansion formulas

rf,

x r+rn+2'f's+k+2

[q-j,,,(qj,aR,jM' ] "(qj+l jer, ql-j jer, bs ' eK ' q, wq ,

(3.7.8)

where we used a contracted notation analogous to that used in (3.7.1). Note that by letting er ----t 00 in (3.7.8) and setting m = 2, II = h = 0 we get the expansion rf, [aR,c r+t'f's+u b d T ; q, xw ]

s' u

X

r+2¢s+k [q-Jb',"(qj,aR;q,w q] ,

(3.7.9)

s,e K

which is equivalent to Verma's [1966] q-extension of the Fields and Wimp expansion (3.7.1). Other types of expansions are given in Fields and Ismail [1975]. AI-Salam and Verma [1984] used the b ----t 0 limit case of the summation formula (3.6.22) to show that Euler's transformation formula 00

Lanbnxn n=O

00

=

k

L(-l)k~! j(k) (x)b.kao,

(3.7.10)

k=O

where and

has the bibasic extension 00 00 k (1 n n) n A L AnBn(xw)n = L(apq\ph-l Xk L ( .) ap ~a )n n=O k=O n=O q, q k-n pq ,p n

v:..

87

3.7 Bibasic expansion formulas

m

oo(kk) . ap q ; p j B. (_)j () J+k x q . j=O q; q j

X "

~

(3.7.11)

The p = q case of (3.7.11) is due to Jackson [1910a]. In order to employ (3.6.22) to extend (3.7.10), replace n in (3.6.22) by j, multiply both sides by Bn+jx n +j (a/b)jqj2, sum from j = 0 to 00, change the order of summation and then replace k by k - nand j by j + k - n to obtain

(3.7.12) Next we replace a by apn+lqn, b by bpn+lq-n, multiply both sides by Anwn and then sum from n = 0 to 00 to get 00

"A B ( )n ~ n n XW n=O

=" 00

k

(

b

-k

)

apq ,pq ;p k-l k (a k/b') X k=O q, q k ~

k (1 - apnqn) (1 - bpnq-n)(aqk /b; q)n

X X

L n=O

() q; q k-n (apq k ,bpq - k;p) n

n An w

(apkqk,bpkq-k;p)j (_'!.-q2n)j+k-n b

~ (q; q)j(aq2k+l /b; q)j 00

x Bj+k( _x)jq(k-n)(j+k-n-l)+(~)+e+k2n+l). (3.7.13) This formula tends directly to (3.7.11) as b ---+ O. By replacing An, B n , x, w by suitable multiples, we may change (3.7.13) to an equivalent form which tends to (3.7.11) as b ---+

00.

In addition, by replacing An, B n , x, w by Anq2(;) ,

B n q-2(;) , bx/a, aw/b, respectively, we can write (3.7.13) in the simpler looking equivalent form 00 00 ( k b -k ) k+l " A B n n _ " apq ,pq ;p k-l (_ )k ( 2 ) ~ n nX W - ~ ( k/b) x q n=O k=O q, aq ;q k

X

k (1 - apnqn) (1 - bpnq-n)(q-k, aqk /b; q)n An wn (apqk, bpq-k;P)n

~

x" 00

~ j=O

'

(apkqk bpkq-k. p) , j B· j j (q, aq 2k+l/b') J+k X q . ,q J.

(3.7.14)

As in the derivation of (3.7.6), one may extend (3.7.14) by replacing Bj+k by Bj+kCn,j+k-n with Cn,o = 1 for n = 0,1, .... Multivariable expansions, which are really special cases of (3.7.6) and (3.7.14), may be obtained by

88

Additional summation, transformation, and expansion formulas

replacing An and Bn in (3.7.6) and (3.7.14) by multiple power series, see, e.g. Gasper [1989a], Ex. 3.22 and, in the hypergeometric limit case, Luke [1969]. For a multivariable special case ofthe AI-Salam and Verma expansion (3.7.11), see Srivastava [1984].

3.8 Quadratic, cubic, and quartic summation and transformation formulas By setting P = qj or q = p1, j = 2,3, ... , in the bibasic summation formulas of §3.7 and using summation and transformation formulas for basic hypergeometric series, one can derive families of quadratic, cubic, etc. summation, transformation and expansion formulas. To illustrate this we shall derive a quadratic transformation formula containing five arbitrary parameters by starting with the q = p2 case of (3.6.16)

n (1 - adp3k) (1 - b/dpk) (a, b;ph(p-2n, ad2p2n /b; p 2h 2k ~ (1 - ad)(l - bid) (dp2, adp2 /b;p2)dadp2n+l, bpl-2n /d;p)k p

(1 - d)(l - ad/b)(l - adp2n)(1 - dp2n /b) (1 - ad)(l - d/b)(l - dp2n)(1 - adp2n /b)'

(3.8.1)

where n = 0,1, .... Change p to q and d to c in (3.8.1), multiply both sides by

(ac 2/b; q2)n(c/b; q)zn C (q2; q2)n(acq; q)zn n and sum over n to get

(3.8.2) Setting

c _

(1 - ac2q4n /b)(d, e, j; q2)n(a 2q3/deJ)n n - (1 - ac2/b)(ac 2q2/bd, ac2q2 /be, ac2q2 /bj; q2)n'

it follows from (3.8.2) that

lOVVg(ac2/b;ac/b,c,cq/b,cq2/b,d,e,j;q2,a2c2q3/dej)

89

3.8 Quadratic, cubic, and quartic formulas

x

(d,e,J;q2)k(a 2c3q4jbde!)k q(~) (ac 2q2jbd,ac2q2jbe, ac 2q2jbJ; q2)k

x 8 W 7(ac 2q4k jb; cl jb, cqk+l jb, dq2k, eq2k, J q2\ q2, a 2c2q3 jde!). (3.8.3)

If we now assume that

a 2c2q = deJ,

(3.8.4)

then we can apply (2.11.7) to get 8W 7(ac 2q4kjb; cqkjb, cqk+ljb, dq2k, eq2k, J q2k; q2,q2)

(ac 2q4k+2jb, bJjac2q2k,abq2k+l, acqk+2jd;q2)oo (acq3k+2, acq3k+l, ac 2q2k+2 jbd, ac2q2k+2 jbe; q2)oo X (acqk+2je,acqk+ljd,acqk+lje,ac 2q2jbde;q2)oo

(beJ j ac2, bdJ j ac 2, J j acqk, J j acqk-l; q2)oo

+ bJq-2k(ac2q4k+2 jb, cqk jb, cqk+l jb, dq2k, eq2k; q2)oo ac2(ac2q2k+2jbJ,acq3k+2,acq3k+l,ac2q2k+2jbd;q2)oo X

(Jq2 je, J q2j d, bJq2-2k jac 2, bJqk+l jc, bJqk+2 jc; q2)oo (ac 2q2k+2jbe,beJjac2, bdJjac2, bJ2q2jac 2, Jjacqk, Jjacqk-l ;q2)oo

X 8 W 7(bJ 2 jac2; J q2k,

beJ jac 2, bdJ jac 2, J jacl, J jacl-\ q2, q2), (3.8.5)

which, combined with (3.8.3), gives

lOW9 (ac 2jb;c,d,e,J,acjb,cqjb,cq2jb;q2,q2) (ac 2q2jb,ac2q2jbde,abq,bJjac2;q2)oo(acqjd,acqje;q)oo (ac 2q2jbd, ac2q2 jbe, bdJ jac2, beJ jac2; q2)oo(acq, J jac; q)oo ~ x

(1 - acq3k)(1 - bjcqk)(a, b, cjb; q)k(d, e, J; q2h

2k

~ (1 - ac)(l - bjc) (cq2, acq2 jb, abq; q2)k(acqjd, acqje, acqj J; q)k q

+

bJ(ac2q2jb, d, e, J q2jd, J q2je, bJq2jac2; q2)oo ac2(ac 2q2jbJ,ac2q2jbe,ac2q2jbd, bdJjac2, beJjac2,bJ2q2jac2;q2)oo (bJqjc, cjb; q)oo x --';--------'------::-;-'---c-(acq, J j ac; q)oo ~ (1 - acq3k)(1 - bjcqk) X

~

(1 - ac)(l - bjc)

(a, b; qh(ac 2jbJ, J; q2)k 2k (cq2, acq2 jb; q2h(bJqjc, acqj J; q)k q

x 8 W 7(bJ 2jac2; J q2k, bdJ jac2, beflac2, J jacqk, J jacqk-\ q2, q2).

(3.8.6)

90

Additional summation, transformation, and expansion formulas

The last sum over k in (3.8.6) is ~ (bP lac2 ,j, bdf lac 2, bef lac2 , f lac, fqlac; q2)j(1 - bp q4 j lac2 ) q2j (q2, bfq2 lac2 ,jq2 Id, fq2 Ie, bfq 2 /c, bfqlc; q2)j(1 - bP lac 2 )

f;:o

~

(1 - acq3k)(1 - blcqk) (a, b; q)dac2 Ibfq2j ,jq2j; q2)k 2k ~ (1 - ac)(l - blc) (cq2, acq2/b; q2)k(bfq2J+1 Ic, acql-2j If; q)k q

X

=

~ (bP lac2 , f, bd!lac2 , bef lac2 , !lac, fqlac; q2)j(1 - bp q4 j lac 2 ) q2j

f;:o

(q2, bfq2 lac2 , fq2ld, fq2 Ie, bfq2/c, bfqlc; q2)j(1 - bP lac2 )

(1 - c)(l - aclb)(l - acl fq2j)(1 - bfq2j Ic) . . c(l - ac)(l- blc)(l - fq2) Ic)(l - aclbfq2J)

x

{

x

(a, b; q)00(Jq2j , ac2 Ibfq2j; q2)00 (c,aclb;q2)00(aclfq2j,bfq2jlc;q)00-

I}

(1 - c)(l - aclb)(l - f lac)(l - bf Ic) (1 - clb)(l - ac)(l - f Ic)(l - bf lac) x 10 W g(bf2 lac2 ; f, bdf lac2 , bef lac 2, bf lac, f Ic, fql ac, fq2 lac; q2, q2)

f

+

ac(l- ac)(l - clb)(l - f Ic)(l- bf lac) (a, b; q)oo(J, ac2/bf; q2)00 x-;----c;o---'---'--;;-'-:-:'-'-~_'__;_____'___c~'-:'--;---;-­ (cq2, acq 2 /b; q2)00(acql f, bfqlc; q)oo 00 (bP lac 2 , bdf lac 2 , bef lac2 , f Ic, bf lac; q2)j (1 - bp q j lac 2 ) ( 2: (q2, fq2 Id,jq2 Ie, bfq2 lac, fq 2/c; q2)j (1 - bP lac2 ) 4

X

fq2)j j2 q ab

--

j=O

(3.8.7) by the n lO W

00

case of (3.6.15). Thus,

2 2 2 2 bf(l - c)(l - aclb) g(ac Ib;c,d,e,f,aclb,cqlb,cq Ib;q ,q ) - ac2(1- !l c )(l- bflac)

x

X

=

(ac 2q2/b, d, e, fq2ld, fq2 Ie, bfq2 lac 2 ; q2)00 2 (bf2q2/ac , ac2q2/be,ac2q2/bd,bdflac2, beflac2 ,ac2q2 /bf; q2)00 (bf Ic, cqlb; q)oo (ac,fqlac;q)oo

W. (bP. f bdf bef f 10

9

2 2)

bf fq fq2. ac2' 'ac2'ac2'~'ac'ac'ac2,q ,q

3.8 Quadratic, cubic, and quartic formulas

x

91

(1, ac2/bf, ac 2q2 /b, d, e, f q2/d, f q2/e, bfq2 /ac 2; q2)00 (cq2, acq2/b, ac 2q2/bf,ac 2q2/be, ac 2q2/bd, bdf/ac2, bef/ac2,bf2q2/ac2;q2)00

x ~ (bP /ac 2, bdflac 2, bef /ac 2 , flc, bflac; q2)j

~

x

(q2, f q2/d, f q2/e, bfq2/ac, f q2/c; q2)j

(1- bpq4j /ac 2) ( -fq2)j j2 - q (1 - bj2 /ac 2) ab

(3.8.8)

when (3.8.4) holds. Now observe that since

f

(a, qva, -qva, c, d, e, f; q)j (_ a 2q2 )j q(~) j=O (q, va, -va, aq/c, aq/d, aq/e, aq/ f; q)j cdef _ (aq,aq/ef;q)oo - (aq/e,aq/f;q)oo

by the n

~ 00

3

¢ [aq/cd,e,f. aq ] 2 aq/c,aq/d,q, ef

(3.8.9)

limit case of (2.5.1), the sum over j in (3.8.8) equals

2 2]

(q2,bPq2/ac 2;q2)00 ¢ [f/c,bf/ac,ac2q2/bde. (bf q2/ ac , f q2/ C; q2)00 3 2 f q2/d, f q2/e ,q ,q

.

(3.8.10)

Hence, by setting e = a 2c2q/df in (3.8.8) we obtain the Gasper and Rahman [1990] quadratic transformation formula

lOVVg(ac2/b;f,ac/b,c,cq/b,cq2/b,d,a2c2q/df;q2,q2) (ac2q2/b,bf/ac2,ac/b,c,cq/b,cq2/b,bfq2/ac;q2)00 (bf2 q2/ac2, ac2/bf,ac2q2/bd,dfq/ab,bdf/ac2,abq/d,cq2; q2 )00

+~~~~~~--~~--~~~~~~~~~~

x (1q2/c, bf /c, bfq/c, fq2 / d, dPq/a 2c2, d, a 2c2q/df; q2)00 (acq2/b,f/c, bf/ac, aC,acq, fq/ac, f q2/ac;q2)00

(a, b, cq/b; q)oo (ac, ac/ f, fq/ac; q)oo

x

(f,d,a2c2q/df,bf/ac2,ac2q2/b,fq2/d,df2q/a2c2,q2;q2)00 (bf/ac, f/c, cq2, acq2/b, ac2q2/bd, dfq/ab, bdf/ac2, abq/d;q2 )00

x

3¢2

[

22]

flC,bf/ac,fq/ab f q2/d, dpq/a2c2 ; q ,q

(acq/d, df lac; q)00(ac2q2 /b, abq, bf /ac 2, fq/ab; q2)00 (acq, f lac; q)00(ac 2q2 /bd, dfq/ab, bdf /ac 2, abq/d; q2)00

x

f

k=O

1 - acq3k (a,b,cq/b;q)k(d,f,a 2c2q/df;q2) kqk 1 _ ac (cq2, acq2/b, abq; q2h(acq/d, acq/ f, df lac; q)k· (3.8.11)

92

Additional summation, transformation, and expansion formulas

Note that the first two terms on the left side of (3.8.11) containing the series can be transformed to another pair of 10 W g series by applying the four-term transformation formula (2.12.9). Since the 31>2 series in (3.8.11) is balanced it can be summed by (1.7.2) whenever it terminates. When c = 1 formula (3.8.11) reduces to the quadratic summation formula 10 W g

~ 1 - aq3k (a, b, q/b; q)k(d, j, a 2q/dj; q2)k k ~ 1 - a (q2, aq2/b, abq; q2h(aq/d, aq/ j, dj fa; q)k q

k=O

(aq, j/a, b, q/b; q)=(d,a 2q/dj,jq2/d, dj2q/a 2;q2)= (a/j,jq/a, aq/d,dj/a; q)=(aq2/b,abq, jq/ab,bj/a; q2)=

+~~~~--=-~~~~~~~~~~~~-=--

X 31>2

2 2]

j, bj la, jq/ab [ jq2/d,dj2q/a 2 ;q ,q

(aq, j/a; q)=(aq2/bd, abq/d, bdj/a, djq/ab; q2)= (aq/d, dj fa; q)=(aq2/b, abq, bj la, jq/ab; q2)= .

(3.8.12)

By multiplying both sides of (3.8.11) by (J lac; q)= and then setting j we obtain Rahman's [1993] quadratic transformation formula ~

~

k=O

=

ac

(a; q 2h(1 - aq3k)(d, aq/d; q2)k(b, c, aq/bc; q)k k (q;q)k(1 -a)(aq/d,d;q)k(aq2/b,aq2/c,bcq;q2)k q

_ ( aq 2,bq,cq, aq 2/bc;q2)= - (q, aq2/b, aq2/c, bcq; q2)=

1> 3

2

[b,c,aq/bc. 2 2] dq, aq2/d ,q ,q ,

(3.8.13)

provided d or aq/d is not of the form q-2n, n a nonnegative integer. Also, the case d = q-2n of (3.8.11) gives n 1 - acq3k (a, b, cq/b; q)k(J, a 2c2q2n+1 / j, q-2n; q2)k k 1 - ac (cq2, acq2/b, abq; q2h(acq/ j, Jlacq2n, acq2n+1; q)k q

~

(acq;q)2n(ac 2q2/bj,abq/j;q2)n (acq/j;q)2n(abq,ac 2q2/b;q2)n x 10 W g (ac 2/b; j, ac/b, c, cq/b, cq2/b, a 2c 2q2 n+l / j, q-2n; q2, q2) (3.8.14)

and the case b = cq2n+1 gives 2n 1 - acq3k (d, j, a 2c 2q/dj; q 2h(a, cq2n+l, q-2n; qh k 1 - ac (acq/d, acq/ j, dJlac; q)k(cq2, aql-2n, acq2n+2; q 2h q

~

(acq2,dq/ac, jq/ac, acq2/dj; q2)n (q/ac,acq2/d,acq2/j,djq/ac;q2)n x 10 W g (acq-2n-\ c, d, j, a 2c2q/dj, aq-2n-\ ql-2n, q-2n; q2, q2)

(3.8.15) for n = 0, 1, .... Similarly, the special case n (l-acq4k)(I-b/cq2k) (a,b;qh(q-3n,ac 2q3n/b;q3h 3k (1 - ac)(1 - b/c) (cq3, acq3/b; q3h(acq3n+1, b/cq3n-1; qh q

~

3.8 Quadratic, cubic, and quartic formulas (1 - e)(1 - ae/b)(1 - aeq3n) (1 - eq3n /b) (1 - ae)(1 - e/b)(1 - eq3n)(1 - aeq3n /b)'

=0 12 n

" , ... ,

93 (3.8.16)

of (3.6.16) is used in Gasper and Rahman [1990] to show that Gosper's sum (see Gessel and Stanton [1982]) F. [ 7 6

a, a + 1/2, b, 1 - b, e, (2a + 1)/3 - e, a/2 + 1 .1] 1/2, (2a - b + 3)/3, (2a + b + 2)/3, 3e, 2a + 1 - 3e, a/2'

2 r(e+!)r(e+~)r(~)r(~)

V3 r (2at2) r (2at 3 ) r (3c+;+l) r (3c+i- b) x r (2+2~-3C) r (3+2~-3C) sin ~(b + 1) r (2+2atb-3c) r (3+2a 3b-3c)

(3.8.17)

has a q-analogue of the form ~ 1 _ aeq4k (a,q/a;q)k(ae;q)2k(d,aeq/d;q3)k qk ~ 1 -ae (eq3,a2eq2;q3)k(q;q)2k(aeq/d,d;q)k

k=O (aeq2,aeq3,d/ae,dq/ae,adq,aq,q2/a,dq2/a;q3)00 (q,q2,dq,dq2,a2eq2,eq3,dq/a2e,d/e;q3)00 d(a,q/a,aeq;q)00(q3,d,aeq/d,d 2q2/ae;q3)00 ae(q,d,aeq/d;q)00(eq3,a2eq2,d/e,dq/a2e;q3)00

+~~~--~7-~~~~~--~~~-­

x

die, dq/a2e 3 3] 2 [ 10

9

a qa~ -qa~ b x -x y -y _q-n q-n

a3q2n+3] a~, -a~, aq/b, aq/x, -aq/x, aq/y, -aq/y, _ aq n+l, aqn+l ,q, - bx 2y2

'"

"

""

.

2 2]

_ (a 2q2, a 2q2 /X 2y2; q2)n ¢ [ q-2n, x 2, y2, -aq/b, - aq2/b . - (a 2q2/x 2, a 2q2 /y2; q2)n 5 4 x2y2q-2n /a2, a 2q2/b 2, -aq, -aq2' q ,q

.

(3.10.4)

98

Additional summation, transformation, and expansion formulas

For a nonterminating extension of (3.10.4) see Jain and Verma [1982]. Since the 5¢>4 series on the right side of (3.10.3) is balanced, it can be summed by (1.7.2) whenever it reduces to a 3¢>2 series. Thus, we obtain the summation formulas: a2,aq2,_aq2: -aq/w,q-n w qn - 1 ]

3 series by Watson's formula (2.5.1), formulas (3.10.5) and (3.10.6) are equivalent to the summation formulas 4¢>3

[

a, qa!, -w/qa!, q-n .1

.1

a 2 ,w,-a 2 q

I-n

;

1

q, q

(w/aq, -a!, aq2 /w; q)n

(3.10.9)

(w, -a-!, aq/w; q)n

and

2

(1 + l/b)(l + a"2 qn /b)(a/b , qa"2 /b, l/b; q)n (1 + qn /b)(l + a! /b)(aq/b, a! /b, 1/b2; q)n ' 1

1

(3.10.10)

respectively, which are closer to what one would expect q-analogues of formulas 4.5(1.1) and 4.5(1.2) in Bailey [1935] to look like.

3.10 Transformations of series

99

It is also of interest to note that if we set c2 = aq in (3.10.3), rewrite the il> series on the left side as an 8¢7 series in base q and transform it to a balanced 4¢3 series, we obtain 4¢3

[

a, b, -w /b, q-n ] /b -b -n ; q, q aq , q ,w

_ (w/a,-aq/b;q)n (w,-q/b;q)n

¢ [ 4

3

2 2] (

a,a 2q2/ w 2,aq/b 2,q-2n . a2q2/b2,aql-n/w,aq2-n/w,q,q

,

)

3.10.11

which is a q-analogue of the c = (1 + a)/2 case of Bailey [1935, 4.5(1)]. Using (2.10.4), the left side of (3.10.11) can be transformed to give q-n, a, aq/b 2, -aq/w ] [ 4¢3 aq /b , -aq /b ,aq I-n/ w ; q, q

(3.10.12) This formula was first proved by Singh [1959] and more recently by Askey and Wilson [1985]. The latter authors also wrote it in the form 4¢3

=

a2,b2,c,d ] [ b! a q2, -ab! q2,-cd; q, q 4¢3

[

2b22d2 a, ,c , 2 2] 2b2 d d; q , q , qa , -c , -qc

(3.10.13)

provided that both series terminate. Now that we have the summation formulas (3.10.5)-(3.10.8), we can use them to produce additional transformation formulas. Set r = 3 and al = -a2 = qa!, a3 = -aq/w, bl = -b 2 = a!, b4 = _aqn+1 in (3.10.1). The il> series on the right side can now be summed by (3.10.5) and this leads to the following q-analogue of Bailey [1935, 4.5(2)]

(-aq, aq2 /w, w/aq; q)n (-q, aq/w, w; q)n X5¢4

[

2 2]

aq,aq2,a2q2/b2c2,a2q2/w2,q-2n a2q2/b2,a2q2/c2,aq2-n/w,aq3-n/w;q,q

.

(3.10.14)

Let us now turn to applications of (3.10.2). If we set r = 1, al = _Aqn+l/a, bl = a 2q-n /A, b2 = _aqn+l, z = q in (3.10.2), where A = (qa 2/bcd)2, then the il> series on the right side reduces to a balanced very-well-poised 8¢7 series in base q which can be summed by Jackson's formula (2.6.2). Thus, we obtain the transformation formula a2,_aq2,b2,c2,d2: _Aqn+l/a,q-n il> [ ;q2,q;q -a, a 2q2 /b 2, a 2q2 / c2, a 2q2 / d 2 : a 2q-n / A, _ aqn+1

1

100

Additional summation, transformation, and expansion formulas

(-aq, >..qj a; q)n(>..q2 j a 2; q2)n (-q, >..qja 2; q)n(>..q2; q2)n

This is a q-analogue of Bailey [1935, 4.5(3)]. Similarly, setting r = 3 and choosing the parameters so that the series on the right side of (3.10.2) can be summed by (3.10.7), we get the following q-analogue of Bailey [1935,4.5(4)] [

2 2 d 2 . ->..qnja q-n a2 'aq2 ] " _aq2 b , c , ., 2 a, -a, a 2q2 jb2, a 2q2 j c2, a 2q2 j d 2 : a 2ql-n j >.., _ aq n+l ; q ,q; q

Exercises 3.1 Deduce from (3.10.13) that

provided that both series terminate. Show that this formula is a q-analogue of Gauss' quadratic transformation formula (3.1.7) when the series terminate. 3.2 Using the sum -+. 2 '/'1

(q-n ql-n. qb2. q2 q2) ,

,

"

2 = (b(b 2'..q2)) n q- (n) 2 , ,q

n

prove that

(i)

q)oo = ((Z; .) az, q

00

-+.

2'/'1

(

a, aq; qb2 ; q2,Z 2) , IZI < 1.

101

Exercises

2

a

2 2 2 ]11 22;q,azq,z../b) (q>..2/b, >..2 /ab 2; q)oo' 1

3.10 Show that 8W7 (

->..; q>,,~, -q>,,~, a, b, -b; q, >..jab2)

(>..ja, >"q, ->..q, >../b2, >..q/ab, ->..q/ab;.q)oo , l>..jab2 < l. (>.., >"q/a, ->..q/a, >"q/b, ->..q/b, >../ab ,q)oo Show that this is a q-analogue of the formula 1

F 3

2

1+>..j2,>"/2,

[a,

b

.

1 + >.. - b'

r(>../2)r (1 + A;a) r(l r(l + >..j2)r (A;a) r(l

1]

+ >.. - b)r(>.. - a + >.. - a - b)r(>.. -

2b) Re(>..-a-2b) >0. 2b)'

3.11 Derive Jackson's [1941] product formula 2(Pl (a 2, b2; qa 2b2; q2,

=

4¢3

z)

2¢1 (a 2, b2; qa 2b2; q2,

] a2, b2, ab, -ab [ 2b2 b! b! ; q, z , a ,a q2, -a q2

qz)

Izl < 1, Iql < 1,

and show that it has Clausen's [1828] formula

[2a, 2b, a + b 1 ; Z ] , Izl < 1, 2a + 2b, a + b + "2 as a limit case. Additional q-analogues of Clausen's formula are given in §8.8. 1 )] 2 [2F1 ( a, b; a + b + -; z = 2

3 F2

3.12 Prove that IJ? [

wq2-n] b2 c2 . d _q1-n/ w q-2n ' _q2-n " ., . 2 • _q-n,q2-2n/b 2,q2-2n/ c2: _q1-n/d,w,q ,q, b2c2d

= (-l,w/d;q)nd n (-d,w; q)n d2,q2-2n/b2c2,q2-2n/w2,q-n,q1-n [ X 5¢4 q2-2n/b2,q 2-2n/ c, 2 dq1-n/ w, dq2-n/ w ; q ,q

2 2] .

104

Additional summation, transformation, and expansion formulas

3.13 If A = a 4q2 /b 2c2d 2 , show that [

a2 ' _q 2a" d2 . . -Aqn/a ] b2 c2, , q-n 2 2 _a,a2q2/b2,a2q2/c2,a2q2/d2: a2ql-n/A,_aqn+l;q ,q;q

(-aq, Aja; q)n(Aja 2 , Aq2/a; q2)n (-q, A/a 2; q)n(Aq2, A/a; q2)n X lOVVg(A; a,aq, Ab2/a 2 , Ac2/a 2 , Ad2/a2,A2q2n/a2,q-2n;q2,a2q3 /A)

and [

=

2a b2 c2 d 2 . _Aqn-l/a q-n a2 " q2a -q ] " ,. , 2 2 a, -a, a 2q2 /b 2, a 2q2 / c2 , a 2q2 / d 2 : a 2q2-n / A, _aqn+l ; q , q; q (-aq,Ajaq;q)n(A/a 2q2;q2)n(1- Aq2n-l/a) n q

(-q, A/qa 2; q)n(Aq2; q2)n(1- Aq-l/a) x lQVVg (A; aq, aq2, Ab2/a 2, AC2/a 2, Ad2/a 2, A2q2n-2 /a 2, q-2n; q2, a 2l/A).

(Nassrallah and Rahman [1981]) 3.14 Using (3.4.7) show that the q-Bessel function defined in Ex. 1.24 can be expressed as

for Ixl < 2. (Rahman [1987]) 3.16 Show that 1

4¢3

[

axq ] a, -qa"2, b, c 1 ; q, -b-a"2, aq/b, aq/c c

_ (l-xa~)(axq;q)oo () x; q 00

-

J.

5'1-'4

[a~,-qa~,(aq)~,-(aq)~,aq/bc. /

/

/

aq b, aq c, q x, axq

(1 - a~)(aq, aq/bc, axq/b, axq/c; q)oo

+~~~~~~--~~~~~-

(aq/b,aq/c,axq/bc,l/x;q)oo

x

J.

5'1-'4

[xa~,-xqd,x(aq)~,-x(aq)~,aXq/bc axq/b,axq/c,qx,aqx 2

] ;q,q .

'

] q, q

Exercises

105

(cz/ab; q2)00 ( /) ~ n (. 2) 2¢1 a,b;c;q,cz abq = ~anZ , n=O z, q 00 show that

3.17 If

(Singh [1959], Nassrallah [1982]) 3.18 If

(cqz/ab; q2)00 ( / ) ~ n 2¢1 a, b; c; q, cz ab = ~ anz ,show that ( . 2) z,q 00 n=O 2¢1(cq/a, cq/b; cq2; q2, z) 2¢1(a, b; c; q2, cz/ab)

(Singh [1959], Nassrallah [1982]) 3.19 If

(cqz/ab; q2)00 2¢1 (/ ) = ~ n a q, b; c/ q; q, cz /ab ~ anz ,show that ( . 2) z,q 00 n=O 2¢1(cq/a, c/bq; c; q2, z) 2¢1(a, b; c; q2, cz/ab)

(Singh [1959], Nassrallah [1982]) 3.20 Prove that

when max(lpl, Iql, Ibl) formulas

< 1, and extend this to the bibasic transformation

~ 1 - apkqk (a;ph(c/b; qh bk 1- a (q; qh(abp;ph

6 =

=

1 - c ~ (ap;ph(c/b; q)k (bq)k 1- b (q; q)k(abp;p)k

6

1 - c ~ (ap;p)k(cq/b; qh bk 1- abp (q;q)k(abp2;ph

6

= (1 - c)(ap;p)oo

f

(b;ph(cqpk; q)oo (ap)k (1 - b) (abp; p)oo k=O (p; p)k (bqpk; q)oo

when max(lpl, Iql, lapl, Ibl) (Gasper [1989a])


1 (qa,qb;qc;q,z) =

fiOO

rq(c) 1 rq(a)rq(b)27ri

-ioo

r q(a+s)rq(b+s)r(-s)r(1+s)(_z)Sds. rq(c+s)rq(1+s) (4.2.6)

4.3 Analytic continuation of 21>I(a, b; C; q, z) Since the integral in (4.2.2) defines an analytic function of z which is singlevalued when larg( -z)1 < 7r, the right side of (4.2.2) gives the analytic continuation ofthe function represented by the series 21>I(a, b; c; q, z) when Izl < 1. As in the ordinary hypergeometric case, we shall denote this analytic continuation of 21>I(a,b;c;q,z) to the domain larg(-z)1 < 7r again by 21>I(a,b;c;q,z). Barnes [1908] used (4.1.1) to show that if larg( -z)1 < 7r and a, b, c, a - b are not integers, then the analytic continuation for Izl > 1 of the series which defines 2Fl(a, b; C; z) for Izl < 1 is given by the equation

r(c)r(b-a) -a -1 2Fl(a,b;c;z) = r(b)r(c-a)(-z) 2Fl(a,1+a-c; 1+a-b;z ) r(c)r(a - b) -b + r(a)r(c-b)(-z) 2Fl(b,1+b-c; 1+b-a;z-1), (4.3.1) where, as elsewhere in this section, the symbol "=" is used in the sense "is the analytic continuation of". To illustrate the extension of Barnes' method to 21>1 series we shall now give Watson's [1910] derivation of the following q-analogue of (4.3.1):

21>1 (a, b; C; q, z) (b, cl a; q)oo (az, ql az; q)oo ( I I I) = (bl ) ( I ) 21>1 a, aq C; aq b; q, cq abz c, a; q 00 z, q Z; q 00 (a, clb; q)oo (bz, qlbz; q)oo ( I I I) + (c, aIb) ; q 00 (z, qI Z; q)00 21>1 b, bq C; bq a; q, cq abz , provided that larg( -z)1

o.

(4.3.2)

< 7r, c and alb are not integer powers of q, and a, b, z =I=-

First consider the integral

h =

_1-/ 27ri

(ql+s, cq S;q)oo7r(_z)S ds (aqs, bqs; q)oo sin 7rS

(4.3.3)

118

Basic contour integrals

along three line-segments AI, A 2 , B, whose equations are: Al : Im(s) = ml,

A2

:

Im(s) = -m2,

B: Re(s) = -M,

(4.3.4)

where ml,m2,M are large positive constants chosen so that A l ,A2 ,B are at least a distance E > 0 away from each pole and zero of

(ql+s, cqs; q)oo

g(s)

= ( ) aqS, bqS; q

(4.3.5) 00

and it is assumed that AI, A 2 , B are terminated by each other and by the contour of the integral in (4.2.2), i.e. Re(s) = 0 with suitable indentations. From an asymptotic formula for (a; q)oo with q = e- w , W > 0 , due to Littlewood [1907, §12]' it follows that if Re(s) ---. -00 with Is - sol> E for some fixed E > 0 and any zero So of (qS; q)oo, then

Re[log(qS; q)ool

=

W

"2(Re(s))

2

W

+"2 Re(s) + 0(1).

(4.3.6)

This implies that

g(s)

=

0

(I :~ IRe(S)) ,

(4.3.7)

when Re(s) ---. -00 with s bounded away from the zeros and poles of g(s). By using this asymptotic expansion and the method of §4.2 it can be shown that if Izl > Icqjabl, then the value of the integral II in (4.3.3) taken along the contours AI, A 2, B tends to zero as ml, m2, M ---. 00. Hence, by Cauchy's theorem, the value of h, taken along the contour C of §4.2, equals the sum of the residues of the integrand at its poles to the left of C when Izl > Icqjabl. Set 0: = -w-lloga, f3 = -w-llogb so that a = q''',b = qf3. Since the residue of the integrand at -0: - n + 27rimw- l is (

-1 l-n

a

-1 -n

n+l

)

q ,ca q ,q ;q -1 -n (q, q, ba q ; q)oo

00

7rW

-l(

-z

)-a-n n(n+l)/2 q

x exp {2m7riw- l log( -z)} csc (2m7r 2iw- l - 0:7r) , we have 00

m=-oo

x

7rw- l (cja, qja; q)oo ( )-a (bj .) -z a, q, q

2¢1

( j j j) a, aq c; aq b; q, cq abz

00

+ idem (a; b).

(4.3.8)

Thus it remains to evaluate the above sums over m when larg(-z)1 < 7r. Letting c = bin (4.3.8) and using (4.2.2), we find that the analytic continuation of 2¢1(a,b;b;q,z) is 00

m=-oo

l qj x 7rw-( (a, )a;q)oo( -z )-a q,q;q 00

2¢1

(a, aq j b; aq j b; q, q jaz) .

4.4 q-analogues of Barnes' first and second lemmas

119

Since, by the q-binomial theorem, 2rP1

(a, b; b; q, z )

(az; q)oo

= (z;q )

00

and the products converge for all values of z, it follows that 00

m=-oo

(4.3.9)

w(q,q,az, q/az; q)oo 7r(a,q/a,z,q/z;q)oo . Using (4.3.9) in (4.3.8) we finally obtain (4.3.2). 4.4 q-analogues of Barnes' first and second lemmas Assume, as before, that 0< q < 1, and consider the integral (q1-c+s " q1-d+s. q) I _ _1 00 7rq S dS 2 - 27ri -ioo (qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s)'

fiOO

(4.4.1)

where, as usual, the contour of integration runs from -ioo to ioo so that the increasing sequences of poles of the integrand (i.e. c + n, d + n with n = 0,1,2, ... ) lie to the right and the decreasing sequences of poles (i.e. the zeros of (qa+s, qb+ s ; q)oo) lie to the left of the contour. By using Cauchy's theorem as in §4.2 to evaluate this integral as the sum of the residues at the poles c + n, d + n with n = 0,1,2, ... , we find that c (q q1+c-d. q) I 7rq , '00 A.. (a+c b+c. l+c-d. ) 2 -. ( d) (a+c ,q ,q ,q, q sm 7r c q , qb+c.) ,q 00 2'/-'1 q

+ idem (c; d).

(4.4.2)

Applying the formula (2.10.13) to (4.4.2), we get 1

fiOO

(q1-c+s, q1-d+s; q)oo

7rqs ds

(qa+s,qb+s;q)oo sin7r(c-s)sin7r(d-s) qC (q, q1+c-d, qd-c, qa+b+c+d; q) 00 sin 7r(c - d) (qa+c, qa+d, qb+ c, qb+d; q)oo

27ri

-ioo

(4.4.3)

which is Watson's [1910] q-analogue of Barnes' first lemma (4.1.2), as can be seen by rewriting it in terms of q-gamma functions. A q-analogue of Barnes' second lemma (4.1.3) can be derived by proceeding as in Agarwal [1953b]. Set c = nand d = c - a - bin (4.4.3) to obtain 1

27ri

fiOO

(q1-n+s,q1-c+a+b+s;q)00

-ioo

= csc 7r( C X

7rqs ds

(qa+s,qb+s;q)oo sin7rssin7r(c-a-b-s) (q1+a+b-c, qc-a-b, qC, q; q)oo a-

b) -'---,----,----------;----:---'-''''(qa, qb, qc-a, qC-b; q)oo

(_l)n (q(a,qb;)q)nqn(C-a-bl-G) qC;q n

(4.4.4)

120

Basic contour integrals

for n = 0,1,2, .... Next, replace c by d in (4.4.4), multiply both sides by

(_I)n qn (e-c)+G) (qC; q)n / (q, qe; q)n' sum over n and change the order of integration and summation (which is easily justified if Iqe-c+sl < 1) to obtain (ql+a+b-d, qd-a-b, qd, q; q) 00 csc 7f( d - a - b) --'--------;----;----;-----;--;--;---'---''-''(qa,qb,qd-a,qd-b;q)oo

x

(qa, l, qC; qd, qe; q, qd+e-a-b-c)

3¢2

jiOO

1

(ql+s,ql-d+a+b+s;q)oo

=

27fi -ioo (qa+s,qb+s;q)oo x 2¢1 (q-S, qC; qe; q, qe-c+s) ds

=

(qe-c; q)oo ~

jiOO

27f~

-ioo

(qe; q)oo

7fqs sin7fssin7f(d-a-b-s)

(ql+s, qe+s, ql-d+a+b+s; q) 00 (qa+s, qb+ s , qe-c+s; q)oo

7fqs ds x sin 7f s sin 7f (d - a - b + s )'

(4.4.5)

by the q-Gauss formula (1.5.1). Now take c = d. Then the series on the left of (4.4.5) can be summed by the q-Gauss formula to give, after an obvious change in parameters,

jiOO

(ql+S, qd+s, qe+s; q) 00 7fqs ds 27fi -ioo (qa+s, qb+ s , qc+s; q)oo sin 7fS sin 7f( d + s) (q, qd, ql-d, qe-a, qe-b, qe-c; q) 00 d =~7f , (qa, qb, qC, ql+a-d, ql+b-d, ql+c-d; q)oo 1

(4.4.6)

where d + e = 1 + a + b + c, which is Agarwal's q-analogue of Barnes' second lemma. This integral converges if q is so small that Re [s log q - log(sin 7fS sin 7f(d + s ))]

Ib1 ... br q/a1 ... a r +11, then the integral I3 = 2;'i t,:,oo ... ds is equal to the sum of the residues of the integrand at those poles of the integrand which lie on the left of the contour. Set a1 = -w- 1 10g a1. Since the residue ofthe integrand at -a1 -n+27rimw- 1 IS

b a -1 q -n , ... , br a 1-1 q -n i q ) ( qn+1 ,a 1-1 q 1-n ,1 -1 _ _ (+1)/2 1 --'-----;------,-------------,-------,----------'---"oo=7rW (-z) 1 nqn n 1 1 (q, q, a2 a l q-n, ... , a r+1 a l q-n i q)oo x csc(2m7r 2iw- 1 - al7r) exp {2m7riw- 1 10g( -z)},

by proceeding as in the proof of (4.3.2) and using (4.3.9) and (4.5.1) we obtain the expansion

(a2, ... , ar+1, bl/a1, ... , br/a1, a1 Z, q/a1 z i q)oo (b1, ... ,br,a2/a1, ... ,ar+1/al,z,q/Ziq)00 A,

X r+1'i/r

[al,a1q/bl, ... ,alq/br. qb1 ... br ] / / ' q, alq a2,···,alq ar+l za1···ar+1

(4.5.2)

+ idem (ali a2,···, ar+d, where the equality holds in the "is the analytic continuation of" sense. The symbol "idem (ali a2, ... , ar+d" after an expression stands for the sum of the r expressions obtained from the preceding expression by interchanging a1 with each ak, k = 2,3, ... , r + 1.

4.6 Contour integrals representing well-poised series Let us replace a, b, c, d and e in (4.4.6) by a + n, b + n, c + n, d + nand e + 2n, respectively, where (4.6.1) e = 1 + a + b + c - d, and transform the integration variable s to s - n, where n is a non-negative integer. Then we get n+ioo (q1+s-n,qd+s,qe+s+n iq )00 1 7rqs-n ds 27ri n-ioo (qa+s , qb+ s , qc+s i q) 00 sin 7r s sin 7r (d + s)

I

= CSC7r

d

(q,qd+n,q1-d-n,qe-a+n,qe-b+n,qe-c+niq)00 ( b +n,qc+n,qe-a- b,qe-a-c,qe- b ) 00 . qa+n,q -c iq

(4.6.2)

The limits of integration n ± ioo can be replaced by ±ioo because we always indent the contour of integration to separate the increasing and decreasing sequences of poles. Thus, it follows from (4.6.2) that (a b c ) ( q,q d ,qI-d ,e e-a ,qe-b ,qe-c iq ) 00 csc(7rd) q ,q ,q iq n (l-d)n (qa, qb, qC, qe-b-c, qe-c-a, qe-a-b i q)oo (qe-a, qe-b, qe-c i q)n q =

1 fioo (q1+S, qd+s, qe+s i q)oo 7rq(n+l)s(q- Si q)n ds 27ri -ioo (qa+s, qb+ s , qc+s i q)oo sin 7rssin 7r(d + s)(qe+s i q)n'

(4.6.3)

122

Basic contour integrals

(4.6.4) where e = 1 + a + b + c - d, Izl < Iqdl and it is assumed that (4.4.7) holds. Therefore, if we can sum the series on the right of (4.6.4), then we can find a simpler contour integral representing the series on the left. In particular, if we let r = 4,e = 1 +A,al = ql+A/2 = -a2,a3 = qd,a4 = qe, _ q2-d-e+A , b1 -- qA/2 -- -b 2, b3 -- ql-d+A , b4 -- ql-e+A ,Z -

then we get a VWP-balanced 6¢5 series which can be summed by (2.7.1). This yields Agarwal's [1953b] contour integral representation for a VWP-balanced 8¢7 series: 8¢7

=

[

qA,ql+A/2,_ql+A/2,qa,qb,qc,qd,qe qA/2, _qA/2, ql+A-a, ql+A-b, ql+A-c, ql+A-d, ql+A-e ; q, q

B]

sin 7r(a + b + c - A)

x

7rqS ds

(4.6.5)

--------~~----------~

sin 7rS sin 7r( a + b + c - A

where B

= 2 + 2A -

+ s) ,

a - b - c - d - e, provided Re(B)

Re [s log q -log(sin7rssin7r(a + b+ c - A

> 0 and

+ s))] < o.

If we evaluate the integral in (4.6.5) by considering the residues at the poles of 1/ [sin 7r s sin 7r( a + b + c - A + s ) ] lying to the right of the contour, then we obtain the transformation (2.10.10) of a VWP-balanced 8¢7 series in terms of the sum of two balanced 4¢3 series. In addition, if we replace A, d, e and a by A, A+d-A, A+e-A and A+a-A, respectively, and take A+a+d+e = 1+2A in (4.6.5), then the integral in (4.6.5) remains unchanged. This gives Bailey's transformation formula (2.10.1) between two VWP-balanced 8¢7 series.

4.7 Contour integral analogue of Bailey's formula

123

4.7 A contour integral analogue of Bailey's summation formula By replacing A, a, b, c, d, e in (4.6.5) by a, d, e, f, b, c, respectively, we obtain the formula

s W7 (qa; qb, qC, qd, qe, qf; q, q)

=

sin n (d + e +

f - a)

(ql+a,qd,qe, qf,ql+a-d-e, ql+a-d-f, ql+a-e- f ; q)oo X

~~~~--~----~--~~------~--~~--~--~~--

(q, ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a- f ,ql+a-d-e- f ; q)oo

1

x-

jiOO

2ni

(ql+S, ql+a-b+s, ql+a-c+s; q) 00 (qd+s, qe+s, qf+ s ; q)oo

-ioo

nqs ds

sin ns sin n(d + e +

f -

a + s)'

(4.7.1) provided the series is VWP-balanced, i.e.,

1 + 2a = b + c + d + e +

f.

(4.7.2)

Since 1 + 2(2b - a) = b + (b + c - a) + (b + d - a) + (b + e - a) + (b + f - a) by (4.7.2), it follows that (4.7.2) remains unchanged if we replace a, c, d, e, f by 2b - a, b + c - a, b + d - a, b + e - a, b + f - a, respectively, and keep b unaltered. Then (4.7.1) gives S W7

X

(q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, qb+ f-a; q, q)

) ( q 1+2b-a ,qb+d-a ,qb+e-a ,qb+f-a ,q l+a-d-e ,q l+a-d-f ,q l+a-e-f.,qoo ~------~~~~--~~--~~~~~--~~~~--~------~=

(q,ql+b-a,ql+b-c, ql+b-d,ql+b-e,ql+b-f, qC; q)oo

1

x2ni

.

= SIn nc X

= sin nc

jiOO -ioo

(ql+b-a+s, ql+b-c+s, ql+s; q) 00

nqs ds

(qb+ d- a+s , qb+ e- a+s , qb+ f -a+s; q) 00 sin n s sin n( c - s)

(ql+2b-a, qb+d-a, qb+e-a, qb+f- a ; q)oo """"'--------,----:;-:-;-----:;-.,-;-------,-------'-'''''' (q, q1+ b- a , q1+ b- c , qC; q)oo

(ql+a-d-e,ql+a-d- f ,ql+a-e- f ;q)oo 1 (q1+b-d, q1+b-e , q1+b- f ; q) 00

2ni

jiOO

(q1+s;q)oo

-ioo

(qd+s; q) 00

(ql+a-c+s, ql+a-b+s; q) 00

nqs ds

(qe+s, qf+ s ; q)oo

sinn(a-b+s)sinn(c+b-a-s)'

x

(4.7.3)

where the second integral in (4.7.3) follows from the first by a change of the integration variable s ----; a-b+s. Combining (4.7.1) and (4.7.3) and simplifying, we obtain (ql+a-b, ql+a-c, ql+a-d, ql+a-e, ql+a- f ; q)oo (q1+a, qC, qd, qe, qf ; q) 00 X SW7

b-a

- q

(qa;qb,qc,qd,qe,qf;q,q) ) (q l+b-a ' ql+b-c , ql+b-d , ql+b-e , ql+b- f '. (qX l (ql+2b-a, qb+c-a, qb+d-a, qb+e-a, qb+f-a; q)oo

124

Basic contour integrals X 8W7

(q2b-a; qb, qb+c-a, qb+d-a, qb+e-a, l+f- a ; q, q)

) ( q I+a-d-e ,q I+a-e-f ,q I+a-d-f.,q(X) (q, qC, qb+c-a; q)(X) q) 1 ji(x) (q l+s , ql+a-b+s , ql+a-c+s.'(x)

x2ni

(qd+s, qe+s, qf+ s ; q)(X)

-i(X)

· nq sSIn n (a_ b) d s sin ns sin n(a - b + s) ,

(4.7.4) when (4.7.2) holds. Evaluating the above integral via (4.4.6), we obtain Bailey's summation formula (2.11.7). Agarwal's [1953b] formula 1 2ITi

i(X) (ql+S, q~a+s, _q~a+s, ql+a-b+s, ql+a-c+s; q) (X)

l,~

(

q"+", q1+j "+', _q1+ j "+', qb+., qo+,; q) oc

(ql+a-d+s,ql+a-e+s,ql+a-f+s;q)(X)

x

nqs ds

------~~--~--~--~----~--------~--~--7

(qd+ s , qe+s, qf+ s ; q)(X)

sin ns sin n(a - b + s)

b (q, ql+a-b, qb-a, ql+a-d-e, ql+a-e- f; q) (X) =cscn(a-)

(qb,qc,qd,qe,qf;q)(X)

( q I+a-d- f ,q I+a-c-d ,q I+a-c-e ,q I+a-c- f.,q ) (X)

x

(qb+c-a, qb+d-a, qb+e-a, qb+ f -a; q) (X)

,

(4.7.5)

where 1 + 2a = b + c + d + e + j, follows directly from (2.11.7) by considering the residues of the integrand of the above integral at the poles to the right of the contour, i.e. at s = n, b - a + n with n = 0,1,2, .... Thus (4.7.5) gives an integral analogue of Bailey's summation formula (2.11.7). The integral in (4.7.5) converges if q is so small that Re [s log q - log (sin n s sin n (a - b + s) )1 < 0 on the contour for large

(4.7.6)

lsi-

4.8 Extensions to complex q inside the unit disc The previous basic contour integrals can be extended to complex q inside the unit disc by using suitable contours. For 0 < Iql < 1, let (4.8.1) where WI = -log Iql > 0 and W2 = -Arg q. Thus q = e-(Wl +iW 2). Then a modification of the proof in §4.2 (see Watson [1910]) shows that if 0 < Iql < 1 and Izl < 1, then formula (4.2.2) extends to 2¢1

(a, b; c; q, z) (a, b; q)(X) (q,c;q)(X)

(4.8.2)

4.9 Other types of basic contour integral formulas

125

where C is an upward directed contour parallel to the line Re( s( WI + iW2)) = 0 with indentations, to ensure that the increasing sequence of poles 0, 1,2, ... , of the integrand lie to the right, and the decreasing sequences of poles lie to the left of C. Since the above integral converges if Re[s log( -z) - log (sin 7fs)] < 0 on C for large lsi, i.e., if (4.8.3) it is required that z satisfies (4.8.3) in order for (4.8.2) to hold. This restriction means that the z-plane has a cut in the form of the spiral whose equation in polar coordinates is r = eW1 (J/W2. Analogously, when 0 < Iql < 1, the contours in the q-analogues of Barnes' first and second lemmas given in §4.4 and the contours in the other integrals in §§4.4-4.7 must be replaced by upward directed contours parallel to the line Re(s(wl + iW2)) = 0 with indentations to separate the increasing and decreasing sequences of poles. 4.9 Other types of basic contour integrals

Let q = e- w with

W

> 0 and suppose that

(aIz, ... ,aAz, bl/z, ... ,bB/z;q)oo P () z = ~----------~~----~~~~ (CIZ, ... ,ccz , dl/z, ... ,dD/z;q)oo

(4.9.1)

has only simple poles. During the 1950's Slater [1952c,d, 1955] considered contour integrals of the form 1m

W == 1m(A,B;C,D) = -.

27f~

J-i7r i7r

/

W

P(qS)qmsds

(4.9.2)

/w

with m = 0 or 1. However, here we shall let m be an arbitrary integer. It is assumed that none of the poles of P(qS) lie on the lines Ims = ±7f/w and that the contour of integration runs from -i7f / W to i7f / wand separates the increasing sequences of poles in 11m sl < 7f/w from those that are decreasing. By setting if) = -sw the integral 1m can also be written in the "exponential" form 1m

= ~ J7r 27f

-7r

P (ei(J) eim(Jdf)

(4.9.3)

with suitable indentations, if necessary, in the contour of integration. Similarly, setting z = qS we obtain that (4.9.4) where the contour K is a deformation of the (positively oriented) unit circle so that the poles of 1/ (CIZ, ... , ccz; q)oo lie outside the contour and the origin and poles of 1/(dl/z, ... ,dD/z;q)oo lie inside the contour. Special cases of (4.9.3) and (4.9.4) have been considered by Askey and Roy [1986]. Although each of the above three types of integrals can be used to derive transformation formulas for basic hypergeometric series, we shall prefer to

126

Basic contour integrals

use mainly the contour integrals of the type in (4.9.4) since they are easier to work with, especially when the assumption in Slater [1952c,d, 1966] that 0< q < 1 is replaced by only assuming that Iql < 1, which is the case we wish to consider in the remainder of this chapter. 4.10 General basic contour integral formulas

Our main objective in this section is to see what formulas can be derived by applying Cauchy's theorem to the integrals Im in (4.9.4). Let Iql < 1 and let 15 be a positive number such that 15 -=1= Idjqnl for j = 1,2, ... , D, and 15 -=1= Icjlq-nl for j = 1,2, ... ,0 when n = 0,1,2, .... Also let ON be the circle Izl = blqlN, where N is a positive integer. Then ON does not pass through any of the poles of P(z) and we have that

IF (bqN)

(bqN) m-l I = I (alb, ... ,aAb, bd 15, ... ,bB/b; q)oo I (clb, ... ,ceb, dd b, ... , dD/b; q)oo

x

I (Clb, ... ,ceb,qb/bl, ... ,qb/bB;.q)N

X

bm-llbNq(Ntl)

(bl ... bBqm-l)NI (alb, ... , aAb, qb/d l , ... , qb/d D, q)N d l ··· d D D I

-

B

=

0

(I bl

~~ ~~~:-l IN IbN q(Ntl) ..

D I

-

B

)

.

(4.10.1) Since ON is of length O(lqIN) it follows from (4.10.1) that if D and

> B or if D = B (4.10.2)

then

lim

N--+oo

1

eN

p(z)zm-ldz

=

O.

(4.10.3)

Hence, by applying Cauchy's residue theorem to the region between K and ON for sufficiently large N and letting N ----+ 00, we find that if D > B or if D = B and (4.10.2) holds, then Im equals the sum of the residues of p(z)zm-l at the poles of 1/ (dd z, ... , dD/ z; q)oo' Therefore, since .

ReSIdue

z

=

(

dqn

1

(d/ z; q)oo

)

=

(_I)ndq2n+G) , (q; q)n (q; q)oo

n

=

0,1,2, ... ,

(4.10.4)

it follows that

(4.10.5)

4.10 General basic contour integral formulas

127

> B, or if D = Band (4.10.2) holds. In addition, by considering the residues of p(z)zm-l outside of K or by just using the inversion z -+ Z-l and renaming the parameters, we obtain if D

1m= (blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo_m ~ (q, dlCl, ... , dDCl, c2/cl, ... , Cc ICl; q)oo

f

x

(dlCl, ... , dDCl, qcI/al, ... , qcI/aA; q)n n=O (q, blCl, ... , bBCl, qCI/C2, ... , qcI/cc; q)n

(n+l)/2)n(C-A) (a l ... aAq-m)n x ( -Clq Cl··· Cc + idem (Cl; C2, ... , cc)

if C

> A, or if C

=

A and l

In the special case when C in the form

(A

Im

(4.10.6)

=

al ... aAq-ml Cl··· Cc

< 1.

A we can use the

reps

(4.10.7) notation to write (4.10.5)

·A ) - (aldl, ... ,aAdl,bI/dl, ... ,bBldl;q)oo d m , B, , D - (q,cl dl , ... ,CAdl ,d2Idl, ... ,dD I ) 00 1 dl;q

cldl , ... , cAdI, qdI/bl , ... , qdI/bB . t( d )D-B] al d 1,···,aA d l,q did 1 2,···,q did 1 D ,q, q 1 + idem (d l ;d2, ... ,dD ) (4.10.8) rI,

[

XA+B'I'A+D-l

where t

= bl b2 ··· bBqm Id l ... d D , if D > B, or if D = Band (4.10.2) holds.

Similarly, from the D

1 (A B·C B) m

=

B case of (4.10.6) we have

(blCl, ... ,bBcl,aI/cl, ... ,aAlcl;q)oo -m (q,dlCl, ... ,dBCl,C2/cl, ... ,cclcl;q)ooCl

=

",

[dlCl, ... , dBCl, qCI/al, ... , qcI/aA. ( )C-A] I I ' q, U qCl blCl, ... ,bBcl,qcl C2,···,qCl Cc + idem (Cl; C2, ... , cc) (4.10.9) rI,

XA+B'I'B+C-l

> A, or if C = A and (4.10.7) holds, where u = al··· aAq-m ICI ... CC. Evaluations of Im which follow from these formulas will be considered in §4.11. From (4.10.8) and (4.10.9) it follows that if C = A and D = B, then we have the transformation formula

if C

(aldl , ... , aAdl , bI/dl , ... , bBldl ; q)oo dm (cldl , ... , cAdI, d2/dl, ... , dBI dl ; q)oo 1 rI,

XA+B'I'A+B-l

[Cldl, ... ,CAdl,qdI/bl, ... ,qdI/bB. bl ... bBqm ] al d 1,···aA d l,q did 1 2,···,qdid 1 B ,q, d 1··· d B

+ idem (d l ; d 2 , ... , dB) =

(blCl, ... , bBCl, aI/cl, ... , aAlcl; q)oo -m ~ (dlCl, ... , dBCl, c2/cl, ... , cAlcl; q)oo

128

Basic contour integrals

'" [dICI, ... ,dBCI,qcI/al, ... ,qcI/aA. al ... a Aq - m ] XA+B 'f'A+B-I bICI,···, bBCI,qCI / C2,···,qCI / CA' q, CI···CA + idem (CI; C2, ... , CA) (4.10.10) provided that Ibl ·· ·bBqml < Idl ·· ·dBI, lal·· ·aAq-ml < ICI·· ·cAI and m = 0,±1,±2, .... In some applications it is useful to have a variable Z in the argument of the series which is independent of the parameters in the series. This can be accomplished by replacing A by A + 1, B by B + 1 and setting bB + I = Z and aA+I = q/z in (4.10.10). More generally, doing this to the m = 0 case of (4.10.5) and of (4.10.6) gives the rather general transformation formula

(aldl , ... , aAdl , bI/dl , ... , bB/dl , z/dl , qdI/ z; q)(XJ (cld l , ... , CedI, d 2/d l , ... , dD/d l ; q)(XJ

x

f

(cldl , ... , CedI, qdI/bl , ... , qdI/bB ; q)n

n=O (q, aldl , ... , aAdl , qdI/d2, ... , qdI/dD; q)n

x (_d l q(n+I J/2)n(D-B-IJ (bl ... bBZ) n dl

··

·dD

+ idem (d l ; d2, ... , d D ) (bl CI, ... , bBCI, aI/ CI, ... , aA/ CI, CIZ, q/ CIZ; q)(XJ (dICI, ... , dDCI, C2/CI, ... , Ce /CI; q)(XJ

x

f

(dICI, ... , dDCI, qCI/al, .. . , qcI/aA; q)n n=O (q, bICI, ... , bBCI, qcI/c2, ... , qcI/ce; q)n

(_Clq(n+I J/2)n(e-A-IJ (a l ... aAq)n CI··· CeZ + idem (CI; C2, ... , ce), X

(4.10.11)

where, for convergence,

(i)

D >B

+ 1,

or D

bl ... bBz I

= B + 1 and I d l

...

dD

A + 1, or C = A + 1 and l

al .. . aAql CI··· CeZ


0, Re d beta integral

_1_liOO 27ri

> 0, and Re(x + y) > 1. Show that Cauchy's [1825]

(1

-ioo

ds = r(x + y - 1)(1 + d/e)l- Y + C/d)l-x (1 + es)x(1- ds)Y (e + d)r(x)r(y)

has a q-analogue of the form

liOO

(-esqX,dsqY;q)oo ds (-es, ds; q)oo r q(x + y - 1) (_eqX /d, -dqY /e; q)oo r q(x)r q(Y) (e + d)( -eq/d, -dq/e; q)oo'

1 27ri

-ioo

°

where < q < 1. (Wilson [1985]) 4.2 Prove that

1 27ri

liOO

(q1+s, _qa+s, qa-b+1+s, _qa-b+1+s, qc+d+e-a+s; q) 00

(qc+s, qd+s, qe+s, _qa+1+s, _qa-2b+s; q)oo 7rqs ds x ~--~--~~~------~ sin 7r s sin 7r (e + d + e - a + s) (q,qc+d+e-a, q1+a-c-d-e,q1+a-b; q)oo =csc7r(e+d+e-a) (q1+ a, _q 1+ a, _qa -2b , qC; q ) 00 -ioo

(_q1+a-b, _qa, q1+a-c, q1+a-d, q1+a-e; q) 00 X

~~--~----~~------~--~--~=

XlO

(qd,qe,q1+a-c-d, q1+a-c-e, q1+a-d-e; q)oo

W g (qa; i q1+a/2, _i q1+a/2, l, _qb, qC, qd, qe; q, _q1+2a-2b-C-d-e) ,

where 1 + 2a - 2b

> e + d + e.

4.3 Show that

ae, be, ad ] 3cP2 [ b dh;q,gh a eg,ae (q, ae, be, ag, bg, eh; q)oo

133

Exercises 4.4 Prove that ~ 27r

j1l"

(Je iO , ke iO Id, qde- iO Ik, cke- iO , qe iO Ick, abcdghe iO I j; q) 00 dB (ae iO be iO ce- iO de- iO ge iO heiO . q)

-7r

" " "

00

(k,qlk,ckld,qdlck,cj,dj,acdg, bcdg, cdgh,abcdhlj; q)oo (q, ac, ad, bc, bd, cg, dg, ch, dh, cdjg; q)oo x 8 W 7 (cdjglq; cg, dg, j la, jib, j Ih; q, abcdhl1)·

4.5 Prove that

[

.,.f...

4'1-'3

q-n , abcdqn-l " ae iO ae- iO ab, ac, ad 2

(q;q)oo (q!, q!, ab, ac, bc; q)oo

1l" (q! eiHi I a, q! ae-i(}-i , aq! eiO - i , q! ei-iO I a, abcei I a; q) 00

1 X

27r { }

x

-7[

(ae i la bei la ce i la ae iO - i ae- iO - i. q) 00 " ' "

(dae-i,bc;q)n (~eird¢ (abce i la, ad; q) n a

and, more generally, (q,az,alz,bz, blz,cz, clz;q)oo (t;;, ql t;;, t;; Z 2, ql t;; Z 2, ab, ac, bc; q)oo 1 {1l" (t;;ze i la, qae- i It;;z, t;;aze- i , qei It;;az, abce i la; q)oo x 27r ) -1l" X

for n

(ae i I a, be i I a, cei I a, aze- i, ae- i I Z; q)oo

(daci,bc;q) (a .",)n n -e''I' d¢ i (abce la, ad; q)n a

= 0,1,2, ... , where

z

=

ei()

and

t;;

is an arbitrary parameter.

4.6 Prove that

(aqle,aqlj,aqlg,aqlh,qlae,qlaj,qlag,qlah;q)oo (qa2, ab, ac, ad, bla, cia, dla; q)oo x lOW9 (a 2; ab, ac, ad, ae, aj, ag, ah; q, q3labcdejgh)

-1

a

+ idem (a; b, c, d) = 0, where Iq31

< labcdejghl.

134

Basic contour integrals

4.7 Prove that _l(alq/bl,alq/b2, ... ,alq/br,q/albl, ... ,q/albr;q)OO (qai, ala2,···, alan a2/al, ... , ar/al; q)oo

al

X

2r+2 W 2r +l (ai; ala2, ... , alan alb l , ... , alb r ; q, qr-l /al ... arb l ... br )

+ idem (al; a2,···, ar) = 0, where r

1,2, ... , and Iqr-ll

=

< lal ... arb l ... brl.

4.8 Show that _ 1 lioo (_csqn+l ' bds , as·, q) oods 27ri -ioo (-CS, ds, basqn-l; q)oo (-a/ c, -bd/ c; q)oo (b, a/ d; q)n (C + d)( -dq/c, -ba/cq; q)oo (q, -cq/d; q)n'

°

where Re(c, d, ba) > and n = 0,1, .... Show that the q-Cauchy beta integral in Ex. 4.1 follows from this formula by letting n ----+ 00 and then setting b = qY, a = _cqx. 4.9 Extend the integral in Ex. 4.8 to 1 lioo (-acs, ac2s/a, ac 2s/ (3, ac2s/'Y, ac2S/6, ac2s/ A; q)oo 27ri -ioo (-cs, as, (3s, "Is, 6s, AS; q)oo

x

(1- a2~s2)

ds

(a/ q, -ac/ a, -ac/ (3, -ach, -ac/6, -ac/ A; q)oo c(q, -a/c, -(3/c, -'Y/c, -6/c, -A/C; q)oo (q2/a , ac2/a(3, ac2/a'Y, ac2/(3"1; q)n X ~~7---~~----~~~~~~~(-cq/a, -cq/ (3, -cqh, -a 2c3 /a(3'Yq; q)n'

where Re(c,a,(3,'Y,6,A) 1,2 ....

>

0, a 3 c5 = -a(3'Y6Aq2, ac = -Aqn+l, and n

4.10 Show that

r

~ (q2z/aa'Y,q2z/ba'Y,q2z/ca'Y,'Y/z;q)oo 1- qz2/a'Y dz 27r2 ) K (az, bz, cz, a/ z; q)oo z

(a'Y, b'Y, c'Y, aqh; q)oo (aa, ba, ca, q; q)oo '

where q2

=

abcaq2, I'Y/al

< 1, and the contour K is as defined in §4.9.

=

Notes 4.11 Show that ~

r (bqz,qzh,'"'(lz;q)oo

JK

27r~

135

(az, bz, al z; q)oo

dz x (qzhqm; q)m(alz; q)ml ... (arz; q)mr Z

b la, aqh, bqla; q)oo (aa, qlaa, ba; q)oo x (qlb'"'(qm; q)m(al/b; q)ml ... (arlb; q)mr (ba)m+ml +.+m r ,

provided

1'"'(1 a < 1, where 1

m, ml, ... ,mr are nonnegative integers, q =

a'"'(qm+ml +·+m r and K is as defined in §4.9. 4.12 Show that 1

-. 27r~

jiOO -ioo

(-acs, dqs; q)oo ( ) ( ) ( d .) al s; q ml . .. ars; q mr ds -cs, S, q 00

(-acld;q)oo

(I)

(I)

(c + d)( -cqld; q)oo al d; q ml ... ar d; q mr provided laq-(m1+·+mr)1 are nonnegative integers.

< 1, Re (c,d,al,···,a r ) > 0, and ml, ... ,mr

4.13 Show that

jiOO

(-acs,-ac 2slf,ac2sla,ac 2sl/3;q)00 27ri -ioo (-cs, - f s, as, /3s; q)oo 2 2 x (ac sh ,ac s15;q)00(1 - ac2 s 21 q) ds ('"'(s, 5s; q)oo (alq, acl f, -acla, -acl/3, -ach, -acI5; q)oo c(q, f Ic, -alc, -/3lc, -'"'( Ic, -5Ic; q)oo (q2 la, ac2la/3, ac2la,",(, ac21/3'"'(; q)n x ~~~~~~~-7~~~~~~~ (-cqla, -cql/3, -cqh, -a2c3 Ia/3'"'(q; q)n (ac 2I Pq, acl f, -ac2I fa, -ac2I f/3, -ac2I f'"'(, -ac2I j8; q)oo +~~~~~~~~~~~~~~~~~~~~~ f(q, cl f, -al f, -/31 f, -'"'(If, -51 f; q)oo (P q2lac 2, ac2/a/3, ac2/a'"'(, ac21/3'"'(; q)n x ~~~--~~~~~~~~~~~~ (- fqla, - fql /3, - fqh, -a2c4 I fa/3,",(q; q)n provided ac = f qn+l, a3 c5 = fa/3'"'(5q2, Re (c, f, a, /3, ,",(, 5) > 0, the integrand has only simple poles, and n = 0,1, .... 1

(For the formulas in Exercises 4.8 - 4.13, and related formulas, see Gasper [1989c].) Notes

§4.4 Kalnins and Miller [1988, 1989] exploited symmetry (recurrence relation) techniques similar to those used by Nikiforov and Suslov [1986], Nikiforov,

136

Basic contour integrals

Suslov and Uvarov [1991]' and Nikiforov and Uvarov [1988] to give another proof of (4.4.3) and of (4.11.1). §4.6 Contour integrals of the types considered in this section were used by Agarwal [1953c] to give simple proofs of the two-term and three-term transformation formulas for 8¢7 series. §4.12 Sears [1951b] also derived the hypergeometric limit cases of the transformation formulas in this section. Applications of (4.12.1) to some formulas in partition theory are given in M. Jackson [1949]. Exercises 4.1-4.5 For additional q-beta integrals, see Askey [1988a,b, 198ge], Gasper [1989c], and Rahman and Suslov [1994a,b, 1996a, 1998].

5 BILATERAL BASIC HYPERGEOMETRIC SERIES

5.1 Notations and definitions The general bilateral basic hypergeometric series in base q with r numerator and s denominator parameters is defined by

r1/Js(z)

== r1/Js [ab1,ab2""'bar;q,z]

f

1, 2,···,

s

(a1, a2,···, ar ; q)n (_I)(s-rlnq(s-rlG) zn. n=-oo (b 1, b2,···, bs; q)n

(5.1.1)

In (5.1.1) it is assumed that q, z and the parameters are such that each term of the series is well-defined (i.e., the denominator factors are never zero, q =I=- 0 if s < r, and z =I=- 0 if negative powers of z occur). Note that a bilateral basic hypergeometric series is a series L~=-oo Vn such that Vo = 1 and vn+dvn is a rational function of qn. By applying (1.2.28) to the terms with negative n, we obtain that

r1/Js(z) =

f

(a1,a2, ... ,ar ;q)n (_I)(s-r ln q(s-rlG)zn ... ,bs;q)n n=O (b1,b2,

+

f

(5.1.2)

(q/b1, q/b2, ... , q/bs;.q)n (b 1'" bs ) n n=l (q/a1, q/a2,"" q/ar , q)n a1 ... arz

Let R = Ib 1 ··· bs /a1'" arl. If s < rand Iql < 1, then the first series on the right side of (5.1.2) diverges for z =I=- 0; if s < rand Iql > 1, then the first series converges for Izl < R and the second series converges for all z =I=- O. When r < sand Iql < 1 the first series converges for all z, but the second series converges only when Izl > R. If r < sand Iql > 1, the second series diverges for all z =I=- O. If r = s, which is the most important case, and Iql < 1, the first series converges when Izl < 1 and the second when Izl > R; on the other hand, if Iql > 1 the second series converges when Izl > 1 and the first when Izl < R. We shall assume throughout this chapter that Iql < 1, so that the region of convergence of the bilateral series

(5.1.3)

137

138

Bilateral basic hypergeometric series

is the annulus (5.1.4) When bj = q for some j, the second series on the right sides of (5.1.2) and (5.1.3) vanish and the first series become basic hypergeometric series. If we replace the index of summation n in (5.1.1) by k + n, where k is an integer, then it follows that

= (al, ... ,ar;q)k z k [(_I)kq(~)]s-r (b l

X

, ... ,

r'l/Js [

bs ; qh

k(S-r J] al qk, ... , arqk k k ; q, zq . blq , ... ,bsq

(5.1.5)

When rand s are small we shall frequently use the single-line notation

r'l/Js(z) == r'l/Js(al,"" ar ; bl , ... , bs; q, z). An r'l/Jr series will be called well-poised if alb l = a2b2 = ... = arbn and verywell-poised if it is well-poised and al = -a2 = qb l = -qb 2. Corresponding to the definition of a VWP-balanced r+l¢r series in §2.1, we call a very-wellpoised r'l/Jr series very-well-poised-balanced (VWP-balanced) if (5.1.6) with either the plus or minus sign. The very-well-poised bilateral basic hypergeometric series in (5.3.1), (5.5.2), (5.5.3), and in §5.6 are VWP-balanced. Note 1

that if in (5.1.6) we replace r by r+l, al by qar, and then a3 by aI, then (5.1.6) reduces to the condition (2.1.12) for a r+l Wr series to be VWP-balanced. A well-poised r'l/Jr series will be called well-poised-balanced (WP-balanced) if

(ala2'" ar)z = -(±(alb1)br

(5.1.7)

with either the plus or minus sign. The well-poised bilateral basic hypergeometric series in (5.3.3), (5.3.4), (5.5.1), (5.5.4), and in (5.5.5) are WP-balanced. 5.2 Ramanujan's sum for l'l/Jl(a;b;q,z)

The bilateral summation formula oj,

1 '1-'1

('b' )_ (q,bja,az,qjaz;q)cxo a, ,q, z - ( b,qj a,z,b j az;q ) cxo ,

Ibjal
7

157

Thus

I(a,b,e,d) bd(1 - q)3(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo X

I

b

d qU

a

(qu/a, qu/b, u; q)oo (u/ab;q)oo

Id dqV (qv/e,v(-v/ed,vq/u,u/v,uv;q)oo qv/d, -v, -q! /u, -q! Iv; q)oo c

1

X

lv:~'2 dqt (t'!~ /u, ~q~ lv, qt; q)oo uq

(tq'2, -tq'2, -qt/uv; q)oo n(a- 1 , b- 1 , -e- 1 , -d- 1 ; q)oo bd(1 - q)2(q; q)'!xo(a/b, bq/a, ab, e/d, dq/e, cd; q)oo X

I

a

b

2

Id dqV (qv(v,uv,-v/ed;q)oo / qv / d, -uv; q)oo n( a-l, b-l, q! , -q! , q)oo Ib d (qu/ a, qu/b, edu; q)oo

d

(qu/ a, qu/b; q)oo qU (-u,u/ab;q)oo

e,

c

-1;

=

b(1 - q)(q; q)~(a/b, bq/a, ab, e, d, cd; q)oo

a

qU (eu, du, u/ab; q)oo

n( -1, q! , -q!, abed; q)oo (q, ab, ae, ad, be, bd, cd; q)oo'

(6.2.8)

by repeated applications of (2.10.18). Since (-I,q!,-q!;q)oo = 2(q!,-q!,-q;q)oo = 2, which follows from (6.2.6) by setting t = 1, we get (6.1.1). By analytic continuation, the restrictions on a, b, e, d mentioned above may be removed.

6.3 Integral representations for very-well-poised

81>7

series

Formulas (2.10.18) and (2.10.19) enable us to use the Askey-Wilson q-beta integral (6.1.1) to derive Riemann integral representations for very-well-poised 81>7 series. Let us first set 1

w(x; a, b, e, d)

=

and

J(a, b, e, d, f, g) = where max(lal, Ibl, lei, Idl,

1 1

-1

h(x;g) w(x; a, b, e, d) h(x; f) dx,

If I, Iql) < 1 and 9 is arbitrary.

h(x;g) h(x;d,f)

1

_!h(x;I,-I,q'2,-q'2) (1 - x) 2 h(. b d) x,a, ,e, 2

(6.3.1)

(6.3.2)

Since, by (2.10.18),

(g/d,g/f;q)oo f(1 - q)(q, d/ f, qf /d, fd; q)oo X

i

d

f

d

(qu/ d, qui f, gu; q)oo qU (gu/df; q)ooh(x; u) ,

(6.3.3)

158

The Askey-Wilson q-beta integral

we have

J(

b d f ) (g/d, g/ f; q)oo a, ,e, , ,g = f(l-q)(q,d/f,qf/d,df;q)oo

{f d

Jd

X

qU

(qu/d, qui f, gu; q)oo (gu/df; q)oo

11 ( -1

)

w x; a, b, e, u dx.

(6.3.4)

By (6.1.1),

1 1

-1

W

b )d _ x, a, ,e, u x -

(.

27f(abeu; q)oo . (q, ab, ae, be; q)oo(au, bu, eu; q)oo

(6.3.5)

Substituting this into (6.3.4), we obtain

J(

=

b d f ) a, ,e, , , 9

i

x d

f d qU

27f(g/d, g/ f; q)oo f( 1 - q) (q,. q)200 (d/f ,q f/d ,J, dff ab,ae, b·) e, q 00

(qu/d, qui f, gu, abeu; q)oo ( ). au,bu,eu,gu / df;q 00

(6.3.6)

The parameters in this q-integral are such that (2.10.19) can be applied to obtain

J(

b d f ) a, ,e, "g

=

27f(g/f,fg,abef,bedf,edaf,dabf;q)00 (q,ad,bd,ed,af, bf,ef,df,ab,ae, be, abedf2; q)oo

x 8W7(abedf2q-\af,bf,ef,df,abedfg-\q,g/f),

(6.3.7)

provided Ig/ fl < 1, if the series does not terminate. By virtue of the transformation formula (2.10.1) many different forms of (6.3.7) can be written down. Two particularly useful ones are

J( a, b,e, d , f ,g )

=

(

27f(ag,bg,eg,abed,abef;q)00 ) q, ab, ae, ad, af, be, bd, bf, cd, ef, abeg; q 00

x 8 W 7(abegq-\ ab, ae, be, g/d, g/ f; q, df),

(6.3.8)

which was derived in Nassrallah and Rahman [1985], and

J(

b d f ) a, ,e, "g

=

27f(ag, bg,eg, dg, fg,abedf/g; q)oo (q,ab,ae,ad,af,be,bd,bf,ed,ef,df,g2;q)00 x 8 W 7(lq-\ g/a, g/b, g/e, g/d, g/ f; q, abedfg-1). (6.3.9)

If the series in (6.3.9) does not terminate, then we must impose the condition labedfg-11 < 1 so that it converges. Note that, if in (6.3.8) we let 0 < q < 1, a = -b = q!, e = qa+!, d = z, f = _qf3+!, 9 = zq' with Re(a, (3) > -~ and then take the limit q --+ 1-, we obtain, after some simplification,

r

1 a . . _ r(a + (3 + 2) f3 -, 2F1(--y,a+1,a+(3+2,z)-r(a+1)r((3+1)Jo X (I-x) (l-xz) dx.

(6.3.10) This shows that (6.3.8) is a q-analogue of Euler's integral representation (1.11.10).

6.4 Integral representations for very-well-poised

1O¢9

series

159

Another limiting case of (6.3.8) was pointed out in Rahman [1986b]. To derive it, replace a,b,e,d,f,g in (6.3.8) by qa,qb,qc,qd,qf,qg, respectively. Also, replace x in the integral (6.3.2) by cos(t log q), which corresponds to replacing ei () by qit. Now let q ---+ 1- and use (1.10.1) and (1.10.3) to get the formula 00 ~ r(a + it)r(a - it)r(b + it)r(b - it)r(e + it)r(e - it) 47f -00 r(2it)r( -2it)

1

x r(d + it)r(d - it)r(j + it)r(j - it) dt r(g + it)r(g - it) r(a + b)r(a + e)r(a + d)r(a + f)r(b + e)r(b + d) r(a + g)r(b + g)r(e + g) x r(b + f)r(e + d)r(e + f)r(a + b + e + g) r(a + b + e + d)r(a + b + e + f) ~--~~--~~~~~----~~

a+b+e+g-l,~(a+b+e+g+l),a+b,a+e, b+e x 7 F.6 1 2(a + b + e + 9 - 1), e + g, b + g, a + g,

[

9 - j,

9 - d, a

+ b + e + d, a + b + e + j

.,

1] ,

(6.3.11)

> O.

where Re (a, b, e, d, f)

6.4 Integral representations for very-well-poised

1O¢9

series

If we set 9 = abedj in (6.3.7), then the 8 W 7 series collapses to one term with value 1 and so we have the formula

1

h(x;I,-I,q!,-q!,abedf) dx h(x; a, b, e, d, f) VI - x 2 27f(abed,abej,bedj,abdj,aedj;q)00 (q, ab, ae, ad, aj, be, bd, bj, cd, ej, dj; q)oo = go(a, b, e, d, f), say, 1

-1

where (max lal, Ibl, lei, Idl, Ijl, Iql)

10 1 x a- 1(1 -

(6.4.1)

< 1. This is a q-analogue of the formula

x)b-1(1 - tx)-a-b dx

=

~~~~(:? (1 -

t)-a, Re(a, b)

> o.

Replace j by jqn in (6.4.1), where n is a nonnegative integer, to get

1

(je i () , je- i (); q)n v(x;a,b,e,d,f)( b dj -1 ace -i() ,abedj ei().,q ) n dx 1

= go

(

(aj, bj, ej, dj; q)n ) a,b,e,d,j (bedj,aedj,abdj,abej;q)n'

(6.4.2)

where

. b d j) v (x, a, ,e, ,

=

(1-

x

2)-! h(x; 1, -1, q!, -q!, abedf) h(. x,a, b,e, d , j)

.

(6.4.3)

160

The Askey-Wilson q-beta integral

Let u

abedj. If

=

1 1 -1

=

Izl < 1, then

(6.4.2) gives the formula

v(x;a,b,e,d,f) r+sWr+4(Ujq-\al, ... ,ar,jei!J,je-i!J;q,z) dx

go(a, b, e, d, f) r+7Wr+6(u jq-\ al, ... , ar, aj, bj, ej, dj; q, z).

(6.4.4)

In particular, for r = 3 and z = u2/ala2a3, we have the formula

1 1 -1

v(x;a,b,e,d,j) 8W7(ujq-\al,a2,a3,jei!J,je-i!J;q,u2/ala2a3) dx

= go(a, b, e, d, f) lOW9(U jq-\ aI, a2, a3, aj, bj, ej, dj; q, u2/ala2a3), (6.4.5) where lu2/ala2a31 < 1, if the series do not terminate. Let us assume that (6.4.6) which ensures that the very-well-poised series on either side of (6.4.5) are balanced. Then, by (2.11.7) 8W 7(u jq-l; aI, a2, a3, jei!J, je-i!J; q, q) (u j, qaI/u j, a2, qaI/a2, a3, qaI/a3; q)ooh(x; j, qaI/ f) (u j /qal, u j /a2, u j /a3, qala2/u j, qala3/u j, q2 aI/u j; q)ooh(x; u, qaI/u)

+~~--~~~~~~~~~~~~~~~~~~~----~

x

8 W 7 (qai!u j;

aI, qala2/u j, qala3/uj, qalei!J /u, qale-i!J /u; q, q) (u j, u j /a2a3, qaI/u j, u / j; q)ooh(x; u/a2, u/a3)

(6.4.7)

and hence

1 1

-1

h(x;I,-I,q!,-q!,u/a2,u/a3) h(x;a,b,e,d,j,u/a2a3)

= go(a, b, e, d, f)

{

dx

viI -x 2

(~j /a2, ~ j /a3, uj ja 2; u / j a3; ~)oo Uj, Uj a2a3, u j, u ja2a3; q

00

x lOW9(U jq-l; aI, a2, a3, aj, bj, ej, dj; q, q)

+

(aj, bj, ej, dj, a2, a3, qaI/a2, qaI/a3; q)oo (u/a, u/b, u/e, u/d, qaal/u, qbal/U, qeal/u,qdal/U; q)oo (qal/aj,qal/bj,qal/ej,qal/dj,qal/uj;q)oo x (u / j, u f!qal, u j /a2a3, u/ ja2a3, q2 aI/u j; q)oo x lOWg(qai!uj; aI, u/ ja2, u/ ja3, qaaI/u, qbaI/u, qeaI/u, qdaI/u; q, q) },

(6.4.8) where u = abedj and ala2 a3q = u 2. Since the integrand on the left side of (6.4.8) is symmetric in a, b, e, d and j, the expression on the right side must have the same property. This provides an alternate proof of Bailey's four-term transformation formula (2.12.9) for VWP-balanced 1O max(lbl, lei, Idl)·

(6.6.1)

If a = ±1, then the functions h(x; ±1) and h(x; a) in the integrand in (6.1.1) cancel and by continuity it follows that

I( ±1, b,e, d) =

27f(±bed; q)oo

(q, ±b, ±e, ±d, be, bd, cd; q)

(6.6.2)

00

However, if lal = 1 and a -I=- ±1, then h(x; a) = 0 for some x in the interval (-1,1) and so the integral in (6.1.1) does not converge. Similarly, this integral does not converge if laqnl = 1 and aqn -I=- ±1 for some positive integer n.

164

The Askey-Wilson q-beta integral If there is a nonnegative integer m such that

(6.6.3) and if ab, ac and ad are not of the form q-n for any nonnegative integer n, then the integral in (6.1.1) converges and we can evaluate it by the following technique. Observe that, since h(x;a)

=

(ae i !:l,ae- i !:l;q)m+1 h(x; aqm+1)

=

a 2m +2qm+m2 h(x; aqm+1, q-m la)lh(x; qla),

(6.6.4)

we have

J( a, b,c, d) 1

=

a -2m-2 q -m-m 1

2

1

x j h(x;1,-1,q2-q2,qla) -1 h(x; b, c, d, aqm+1, q-m fa)

dx

V1- x 2 '

(6.6.5)

where the parameters b, c, d, aqm+1, q-m I a in the denominator of the integrand are now all less than 1 in absolute value. By (6.3.8),

J(

b d) a, ,c,

=

2n(qbla,qcla,qdla,abcdqm+l,bcdq- m l a;q)00 (q, bc, bd, cd, abqm+1, acqm+l, adqm+l, bcdqla; q)oo a- 2m- 2q-m-m 2 x ~--~----~~~--~~-(bq-m la, cq-m la, dq-m la; q)oo

x 8 W 7(bcda-\ bc, bd, cd, q-m a-2, qm+\ q, q).

(6.6.6)

The series in (6.6.6) is balanced and so we can apply Bailey's summation formula (2.11.7). After some simplification we find that 2n(abcd;q)00 ) q,ab,ac,ad,bc,bd,cd;q 00

J( a, b,c, d) = (

+ L m (·b a, ,c, d) ,

(6.6.7)

where L m

. b d _ 2n( aql d, bql d, cql d, qm+1, a 2bcqm+1, q-m I a 2 , bcq-m; q)oo (a, ,c, ) - (q, ab, ac, bc, abqm+1, acqm+1, adqm+l, aqm+l I d, abcql d; q)oo a-2m-1d-1q-m-m2

x

~--~------~~~~----~~~

(bq-m I a, cq-m I a, dq-m I a, q-m I ad; q)oo

x 8 W 7(abcd-\ bc, q-m lad, aqm+1 Id, ab, ac; q, q).

(6.6.8)

By (2.10.1), 8 W 7 (abcd-\

bc, q-m lad, aqm+1Id, ab, ac; q, q)

(abcqld,qlad,abq, acq; q)oo (bqld,cqld,q, a 2bcq; q)oo x 8 W7(a 2 bc; abcd, ab, ac, a 2 qm+1, q-m; q, qlad).

(6.6.9)

6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1

165

Since m is a nonnegative integer, the series on the right side of (6.6.9) terminates and hence, by Watson's formula (2.5.1), 8

2 yy 7 a 2b c,. a bcd ,ab,ac, a 2 qm+1 ,q -m., q, q I a d) -_ ((a bbcq, q; q)m ) a q,acq;q m ¢ [q-m, ab, ac, q-m I ad . ] _ (a 2bcq, q, aqlb, aql c; q)m X43 bcq-m, aqld, q-m,q,q - (qa2,abq,acq,qlbc;q)m q] a 2 , qa, -qa, ab, ac, ad, a 2qm+1, q-m [ X 8¢7 a, -a, aq Ib ,aq I c, aq Id , q-m ,a 2 qm+1; q, ab cd

TXT (

(a 2bcq,q,aqlb,aqlc;q)m (qa 2 ,abq,acq,qlbc;q)m

f

X

(a 2;q)k(l-a 2q2k)(ab,ac,ad;q)k ( q )k (q; qh(1 - a 2 )(aqlb, aqlc, aqld; q)k abcd .

k=O

(6.6.10)

Using (6.6.10) and (6.6.9) in (6.6.8), we obtain L m

2n(a- 2; q)= ( . b d) = _ a, ,c, (q,ab,ac,ad,b I a,c I a,d I a;q )=

f

X

k=O

(a 2; q)k(1 - a 2q2k)(ab, ac, ad; q)k ( q )k (q; qh(1 - a 2 )(aqlb, aqlc, aqld; q)k abcd .

(6.6.11)

Hence

1

1 h(x;I,-I,q!,-q!)

-1

X

f

h(x; a, b, c, d)

k=O

2n(a- 2;q)=

dx

VI -

x2

+

(q, ab, ac, ad, bla, cia, dla; q)=

(a 2;q)k(l-a 2q2k)(ab,ac,ad;q)k ( q )k (q; q)k(1 - a 2)(aqlb, aqlc, aqld; qh abcd

2n(abcd; q)= (q, ab, ac, ad, bc, bd, cd; q)= '

(6.6.12)

where max(lbl, Icl, Idl, Iql) < 1, laqm+ll < 1 < laqml for some nonnegative integer m, and the products ab, ac, ad are not of the form q-n, n = 1,2, .... Askey and Wilson [1985] proved this formula by using contour integration. By continuity, formula (6.6.12) also holds if the restriction (6.6.3) is replaced by aqm = ±l. Note that, if one of the products ab, ac or ad is of the form q-n for some nonnegative integer n, the integral in (6.6.12) converges even though the denominator on the right side of (6.6.12) equals zero as does the denominator in the coefficient of the sum in (6.6.12). If we let ab tend to q-n then, since Ibl < 1 and laqm+11 < 1 < laqml, we must have n:S; m. We may then multiply (6.6.12) by 1 - abqn and take the limit ab ~ q-n. The result is a terminating 6¢5 series on the left side and its sum on the right, giving the summation formula (2.4.2). If max(lcl, Idl, Iql) < 1 and there are nonnegative integers m and r such that (6.6.13)

166

The Askey-Wilson q-beta integral

then the above technique can be extended to evaluate I(a, b, e, d) provided the products ab, ae, ad, be, bd are not of the form q-n and alb i=- q±n for any nonnegative integer n. Splitting h(x; b) in the same way as in (6.6.4) and using (6.3.4) we get

I(a,b,e,d) = b- 2r - 2q-r-r 2 J(a, bqr+1, q-r /b, e, d, q/b) - 2r - 2 -r-r2 q

=b

j

X

d

c

d

qU

(q/be, q/bd; q)oo d(l - q)(q, e/d, dq/e, cd; q)oo

(qu/e, quid, qu/b; q)oo (/b d. ) qu e, q 00

11 (. -1

W

b r+1 -rib ) d x, a, q , q ,u x. (6.6.14)

However, by (6.6.12),

1 w(x; q, bqr+l, q-r /b, u) dx r 1-1

27r( aqu; q)oo (q, q, abqr+1, aq-r /b; q)oo(au, buqr+l, uq-r /b; q)oo 27r(a- 2; q)oo (q, abqr+1, q-r lab, bqr+1/a; q)oo(au, u/a; q)oo X

~ (a 2; q)k(l - a 2q2k)(au; q)k ( )-k ~ (q; qh(l _ a2)(aq/u; q)k au .

(6.6.15)

Since by (2.10.18),

j

d

c

dqu (qu/e, quid, qu/b; q)oo (au; qh (qu)-k (qu/bed, au, u/a; q)oo (aq/u; q)k

= (_l)k -2k -k(k+1)/2

jd

(qu/e, quid, qu/b; q)oo ,uq- k/ a;q ) 00 d(l - q)(q, e/d, dq/e, cd, q/ab, aq/b; q)oo (q/be,q/bd, ae, ad, e/a, d/a; q)oo a

q

c

(ab,ae, ad; qh X (aq/b,aq/e,aq/d;q)k

d

qU ( qu /b e d ,auq k

(q)k abed '

(6.6.16)

we find that

I(a, b, e, d)

+

= Lm(a; b, e, d)

27r(adq, edq, dq/b, qr+2, q1-r /b 2; q)oob- 2r - 2q-r-r 2 (q,q,ad,ed,dq2/b,abqr+1,beqr+1,bdqr+1,aq-r/b,eq-r/b,dq-r/b;q)oo

x 8 W 7(dq/b; bdqr+1, q, dq-r /b, q/be, q/ab; q, ae),

(6.6.17)

where Lm (a; b, e, d) is as defined in (6.6.11). The reduction of this 8 W 7 will be done in two stages. First we use (2.10.1) twice to reduce it to a balanced 8 W 7 and then apply (2.11.7) to obtain 8 W 7(dq/b;bdqr+1,q,dq-r/b,q/be,q/ab;q,ae)

6.6 The Askey-Wilson integral when max (Ial, Ibl, Icl, Idl) ~ 1

167

(dq2/b,q/bd,abcdqr+1,acdq-r/b;q)00 (q1-r/b 2,qr+2,ac,qacd2;q)00 X 8 W7( acd 2; abcd, bdqr+1, dq-r /b, ad, cd; q, q/bd) (dq2/b,aq/b,cq/b,q, abcdqr+1,acdq-r/b; q)oo (ac,adq, cdq, acdq/b,qr+2, q1-r/b 2; q)oo x 8 W 7 (acdb-\ cd, ad, ac, qr+1, q-r /b 2; q, q) (dq2/b, aq/b, bfa, q, q/ab, abcd, bcqr+1, bdqr+1, cq-r /b, dq-r /b; q)oo (ac,adq, cdq,dq/b, bd, bc,qr+2, bqr+1/a, q-r/ab, q1-r/b 2; q)oo + ~ (dq2/b, q, bq/a, cq/a, dq/a, ad, qr+1, b2cdqr+1, cdq-r, q-r /b 2; q)oo a (bcdq/a, dq/b, bc, bd, adq, cdq, qr+2, bqr+1 la, q-r lab, q1-r /b 2; q)oo X 8 W 7 (bcda-\

bc, bd, cd, bqr+1 la, q-r lab; q, q).

(6.6.18)

Substituting (6.6.18) into (6.6.17) and simplifying the coefficients we get

27r(abcd; q)oo I(a, b, c, d) - Lm(a; b, c, d) - ( b d b bd d ) q,a ,ac,a ,c, ,c;q 00 27r(bq/a, cq/a, dq/a, qr+1, b2cdqr+1, cdq-r, q-r/b 2; q)oo (q,bc,bd,cd,abqr+1,bcqr+1,bqr+1/ a,aq-r/b,cq-r/b,dq-r;q)00 b-2r-1 -1 -r-r2 X ( -r / b : dq/ .) 8 W 7 (bcda-\ bc, bd, cd, bqr+1 la, q1-r lab; q, q). a, c q a, q 00 q (6.6.19) The expression on the right side of (6.6.19) is the same as that in (6.6.8) with a, b, c, d, m replaced by b, d, c, a and r, respectively, and so has the value

27r(b- 2; q)oo (q,ba, bc,bd,a/b,c/b,d/b; q)oo (b 2; q)k(l- b2q2k)(ba, bc, bd; q)k ( q )k X (q; q)k(l - b2 )(bq/a, bq/c, bq/d; q)k abcd

t

k=O

= Lr(b; c, d, a),

(6.6.20)

by (6.6.11). So we find that

I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) 27r(abcd; q)oo (q, ab, ac, ad, bc, bd, cd; q)oo'

(6.6.21)

where the parameters satisfy the conditions stated earlier. It is now clear that we can handle the cases of three or all four of the parameters a, b, c, d exceeding 1 in absolute value in exactly the same way. For example, in the extreme case when min(lal, Ibl, lei, Idl) > 1 > Iql with

laqm+11 < 1 < laqml, Icq s+11

< 1 < IcqSI,

Ibqr+11 < 1 < Ibqrl, Idqt+11 < 1 < Idqtl,

(6.6.22)

for some nonnegative integers m, r, s, t such that the pairwise products of a, b, c, d are not of the form q-n and the pairwise ratios of a, b, c, d are not

168

The Askey-Wilson q-beta integral

of the form q±n for n

=

0,1,2, ... , we have the formula

I(a, b, c, d) - Lm(a; b, c, d) - Lr(b; c, d, a) - Ls(c; d, a, b) - Lt(d; a, b, c) 27r( abcd; q)oo (6.6.23) (q, ab, ac, ad, bc, bd, cd; q)oo' where Ls(c; d, a, b) and Lt(d; a, b, c) are the same type of finite series as those in (6.6.11) and (6.6.20), and can be written down by obvious replacement of the parameters.

Exercises

6.1 Prove that

17r

sin 2 () d()

IT

(1- 2aj cos() + a;) j=1 when max(la11, la21, la31, la41) o

2

7r(1 - a1 a2a3a4) I1 (1 - ajak) 1 0 and Re b > (Askey and Roy [1986], Gasper [1987])

1-q rq(a+b)

o.

6.17 Extend the above q- beta integral formulas to

(i)

{OO t c - 1 (-tqb, _qa+l It; q)oo d t

Jo

(-t, -qlt; q)oo q (_qC, _ql-c; q)oo r q(a + c)r q(b - c) (-l,-q;q)oo rq(a+b)

and

(ii)

{OO t c - 1 (_tqb, _qa+~ It; q)oo dt

Jo

(-t, -qlt, q)oo r(c)r(l- c)rq(a + c)rq(b - c) rq(c)rq(l- c)rq(a + b)

where 0 < q < 1, Re (a + c) > 0 and Re (b - c) > o. (See Ramanujan [1915]' Askey and Roy [1986], Gasper [1987], Askey [1988b], and Koornwinder [2003]) 6.18 Use Ex. 2.16(ii) to show that, for y

= cos ¢;,

if x> y, if x < y, if x = y. (Ismail and Rahman [2002b]) 6.19 Denoting S(al' ... ' am) Ex. 2.16(i) to prove that

= S(al' ... ' am; q) =

n:=l (ak, qlak; q)oo,

use

Notes

173

Deduce that n

r

7r

io

rr rr

(2W -2iIJ.) h(cosf); Ak, q/Ak) e ,e ,q 00 k=1 df) h(cos f); a, b) n + l ( f ) /) h cos ;am,q am m=1

n+l n .) S(Ak/aj, Akaj) q, q, q 00 j=1 k=1

21f =

(1 _ ab)(

L II

n+l X

II [S(araj, ar/aj)(aaj, baj, aq/aj, bq/aj; q)oo]-I. r=l r=l-j

(Ismail and Rahman [2002b]) 6.20 Show that

1

1 h(x; 1, -1, ql/2, _ql/2, aql/2, ql/2/ a )

-1

h(x;al,a2, ... ,a6)

dx

viI -x 2

rr (aajql/2, ajql/2 / a; q)oo j=1 (q, q, qa 2, q/ a 2; q)oo rr (ajak; q)oo l::;j 0, Re y > 0, Re z > - min(l/n, Re x/(n - 1), Re y/(n - 1)). Aomoto [1987] considered a generalization of Selberg's integral and utilized the extra freedom that he had in his integral to give a short elegant proof of it. Habsieger [1988] and Kadell [1988b] proved a q-analogue of Selberg's integral that was conjectured in Askey [1980b]. For conjectured multivariable extensions of the integrals in Exercises 6.11-6.13, other conjectured q-analogues of Selberg's integral, and related constant term identities that come from root systems associated with Lie algebras, see Andrews [1986, 1988], Askey [1980b, 1982b, 1985a, 1989f, 1990], Evans, Ismail and Stanton [1982]' Garvan [1990], Garvan and Gonnet [1992]' Habsieger [1988], Kadell [1988a-1994]' Macdonald [1972-1995]' Milne [1985a, 1989], Morris [1982]' Rahman [1986a], Stanton [1986b, 1989], and Zeilberger [1987, 1988, 1990a]. Also see the extension of Ex. 6.11, the q-Selberg integrals, and the identities in Aomoto [1998], Evans [1994]' Ito [2002]' Kaneko [1996-1998], Lassalle and Schlosser [2003], Macdonald [1998a,b], Rains [2003a], Stokman [2002] and Tarasov and Varchenko [1997].

7 APPLICATIONS TO ORTHOGONAL POLYNOMIALS

7.1 Orthogonality Let a(x) be a non-constant, non-decreasing, real-valued bounded function defined on (-00,00) such that its moments /Ln =

I:

xnda(x),

n = 0,1,2, ... ,

(7.1.1)

are finite. A finite or infinite sequence Po (x), Pl (x), ... of polynomials, where Pn(x) is of degree n in x, is said to be orthogonal with respect to the measure da(x) and called an orthogonal system of polynomials if (7.1.2) In view of the definition of a(x) the integrals in (7.1.1) and (7.1.2) exist in the Lebesgue-Stieltjes sense. If a(x) is absolutely continuous and da(x) = w(x)dx, then the orthogonality relation reduces to

I:

Pm(X)Pn(X)W(X) dx = 0,

m -=I- n,

(7.1.3)

and the sequence {Pn (x)} is said to be orthogonal with respect to the weight function w(x). If a( x) is a step function (usually taken to be right-continuous) with jumps Wj at x = Xj,j = 0,1,2, ... , then (7.1.2) reduces to LPm(Xj)Pn(Xj)Wj = 0,

m -=I- n.

(7.1.4)

j

In this case the polynomials are said to be orthogonal with respect to a jump function and are usually referred to as orthogonal polynomials of a discrete variable. Every orthogonal system of real valued polynomials {Pn(x)} satisfies a three-term recurrence relation of the form

(7.1.5) with p-l(X) == O,Po(x) == 1, where An, B n , C n are real and AnCn+ l > 0. Conversely, if (7.1.5) holds for a sequence of polynomials {Pn(x)} such that P-l(X) == O,Po(x) == 1 and An, B n , C n are real with AnCn+ l > 0, then there exists a positive measure da(x) such that (7.1.6) 175

176

Applications to orthogonal polynomials

where

_ rrn A k - l Vn C' k

k=l

Vo = 1.

(7.1.7)

°

If {Pn(X)} = {Pn(x)}~=o and AnCn+1 > for n = 0,1,2, ... , then the measure has infinitely many points of support, (7.1.5) holds for n = 0,1,2, ... , and (7.1.6) holds for m, n = 0,1,2, .... If {Pn(x)} = {Pn(x)};;=o and AnCn+1 > for n = 0,1,2, ... , N -1, where N is a fixed positive integer, then the measure can be taken to have support on N + 1 points xo, Xl,"" XN, (7.1.5) holds for n = 0,1, ... , N - 1, and (7.1.6) holds for m, n = 0,1,2, ... , N. This characterization theorem of orthogonal polynomials is usually attributed to Favard [1935], but it appeared earlier in published works of Perron [1929]' Wintner [1929] and Stone [1932]. For a detailed discussion of this theorem see, for example, Atkinson [1964], Chihara [1978], Freud [1971] and Szego [1975]. In the finite discrete case the recurrence relation (7.1.5) is a discrete analogue of a Sturm-Liouville two-point boundary-value problem with boundary conditions p-I(X) = O,PN+I(X) = 0. If Xo, Xl, ... , XN are the zeros of PN+l(X), which can be easily proved to be real and distinct (see e.g., Atkinson [1964] for a complete proof), then the orthogonality relation (7.1.6) can be written in the form

°

N

N

LPm(Xj)Pn(Xj)Wj = V;;:-l L Wj bm,n, (7.1.8) j=O j=O m, n = 0,1, ... , N, where Wj is the positive jump at Xj and Vn is as defined in (7.1.7). The dual orthogonality relation N

N

LPn(Xj)Pn(Xk)Vn = wjl L Wn bj,k, n=O n=O

(7.1.9)

j, k = 0,1, ... , N, follows from the fact that a matrix that is orthogonal by rows is also orthogonal by columns. It can be shown that

Wj = [ANvNPN(Xj )P~+l (Xj)

r

N

lL

Wn,

j

= 0, 1, ... ,N,

(7.1.10)

n=O

where the prime indicates the first derivative. In general, the measure in (7.1.6) is not unique and, given a recurrence relation, it may not be possible to find an explicit formula for o:(x). Even though the classical orthogonal polynomials, which include the Jacobi polynomials

P~Q,(3)(x) =

(0:

:!I)n

2FI (

-n, n

+ 0: + (3 + 1; 0: + 1; 1; X)

,

(7.1.11)

and the ultraspherical polynomials

C~(x)

((2>')t) p2-~')..-~)(x) >'+"2n n _ L -'(>'h(>')n-k -----'---:----'----'------:-:- ei(n-2k)e

=

-

k=O

k!(n-k)!

'

X

= cos e,

(7.1.12)

7.2 The finite discrete case

177

are orthogonal with respect to unique measures (see Szego [1975]), it is not easy to discover these measures from the corresponding recurrence relations (see e.g., Askey and Ismail [1984]). However, for a wide class of discrete orthogonal polynomials it is possible to use the recurrence relation (7.1.5) and the formulas (7.1.8)-(7.1.10) to compute the jumps Wj and hence the measure. We shall illustrate this in the next section by considering the q-Racah polynomials (Askey and Wilson [1979]).

7.2 The finite discrete case: the q-Racah polynomials and some special cases Suppose {Pn(x)} is a finite discrete orthogonal polynomial sequence which satisfies a three-term recurrence relation of the form (7.1.5) and the orthogonality relations (7.1.8) and (7.1.9) with the weights Wj and the normalization constants Vn given by (7.1.10) and (7.1.7), respectively. We shall now assume, without any loss of generality, that Pn(XO) = 1 for n = 0,1, ... , N. This enables us to rewrite (7.1.5) in the form

(x - XO)Pn(x) where n

=

An [Pn+l(X) - Pn(x)]- en [Pn(x) - Pn-l(X)] ,

= 0,1, ... ,N.

Setting j

= k = 0 in (7.1.9)

N

L Vn =

(7.2.1)

we find that

N

L

(7.2.2) W n· n=O n=O It is clear that in order to obtain solutions of (7.2.1) which are representable W;;l

N

in terms of basic series it would be helpful if Vn and quotients of products of q-shifted factorials. (2.4.2) in mind, let us take

Vn

L

Vn were equal to n=O Therefore, with the 6¢5 sum

(abq; q)n (1 - abq2n+l) (aq, eq, bdq; q)n ( )-n (q; q)n(1- abq) (bq, abq/e, aq/d; q)n edq

=

n

II

=

k=l

(1 _ abqk) (1 - abq2k+l) (1 - aqk)(l - eqk)(l - bdqk) (1- qk) (1- abq2k-l) (1- bqk)(l- abqk/e)(l- aqk/d)edq' (7.2.3)

where bdq

=

q-N, 0 < q < 1, so that

~

[ab q, q(abq)~, -q(abq)~, aq, eq, q-N .

_

~ Vn -

n=O

6¢5

1

1

N

(abq)'i, -(abq)'i, bq, abq/e, abq +

2'

(abq2, b/e; q)N (bq, abq/e; q)N' where it is assumed that a, b, e, d are such that Vn > 0 for n view of (7.1. 7) we can take A

q,

bqN ]

(7.2.4)

= 0,1, ... , N.

_ (1 - abqk) (1 - aqk) (1 - eqk) (1 - bdqk) (1 _ abq2k-l) rk,

k-l -

e

In

(7.2.5)

178

Applications to orthogonal polynomials

(7.2.6) where {rdf=l is an arbitrary sequence with rk =I- 0,1 ::::; k ::::; N. Since Co = 0 and Ao = (1 - aq)(l - eq)(l - bdq)rl' we have from the n = 0 case of (7.2.1) that -1 (1- q-l)(Xj - xo)qr 11 Pl ( X· ) (7.2.7) J (1 - q)(l - aq)(l - eq)(l - bdq) This suggests that we should look for a basic series representation of Pn (x j ) whose (k + l)-th term has (q, aq, eq, bdq; qh as its denominator, which in turn suggests considering a terminating 4¢3 series. In view of the product ( -n) k (n+ a+,8 + l)k in the numerator ofthe (k+ l)-th term in the hypergeometric series representation of p~<x,(3)(x) in (7.1.11) and the dual orthogonality relations (7.1.8) and (7.1.9) required for Pn(Xj) it is natural to look for a 4¢3 series whose (k + l)-th term has the numerator

Replacing q<X, qf3, q'Y, q8 by a, b, e, d, respectively, we are then led to consider a 4¢3 series of the form 4¢3

[

q-n, abqn+l, q-j, edqj+l . ] b d ' q, q . aq, eq, q

(7.2.8)

Observing that (q-j, edqJ+\ q) k is a polynomial of degree k in the variable q-j + edqj+1, we find that if we take

(7.2.9) then Xj - Xo = -(1- q-j)(l- edqJ+l) , and so (7.2.7) is satisfied with

(7.2.10) Then AkCk+1 > 0 for 0 ::::; k ::::; N - 1 if, for example, a, b, e are real, d b-1q-N-1, be < 0, max(laql, Ibql, leql, lablel) < 1 and 0 < q < 1. We shall now verify that _ Pn(Xj) satisfies (7.2.1) with x

4¢3

[q-n,abqn+l,q-j,edqj+l. ] b d ' q, q aq, eq, q

=

(7.2.11)

= Xj. A straightforward calculation gives

(7.2.12)

7.2 The finite discrete case

o< n

179

::::: N. So we need to verify that

q-n, abqn+l, q-j, edqj+l ] b d ; q, q aq, eq, q Anq-n (1 - abq2n+2) [q-n, abqn+2, ql- j , edqj+2 . _ ] - (1 - aq) (1 - eq) (1 - bdq) 4¢3 aq2, eq2, bdq2 ' q, q

4¢3

[

C nq l-n (1- abq2n) - (l-aq)(l-eq)(l-bdq)

4¢3

[ql-n, abqn+l, ql- j , edqj+2 . ] aq2,eq2,bdq2 ,q,q,

(7.2.13) where An and C n are given by (7.2.5) and (7.2.6) with Tk as defined in (7.2.10). Use of (2.10.4) on both sides of (7.2.13) reduces the problem to verifying that [

4¢3

q-n, abqn+l, bq-j Ie, bdqi+l ] bq, a bq I e, bdq ; q, q

(1- abqn+l)

(1- bdqn+l)

(1 - abq2n+l) (1 - bdq) x

4¢3

[

q-n, abqn+2, bq-j Ie, bdqi+l ] bq, a bq I e, bd q2 ; q, q

(1 - qn) (1 - aqn Id) (1 - abq2n+l) (1 _ bdq) bdq X4¢3

[

ql-n, abqn+l, bq-j Ie, bdqi+l ] bq, a blbd2 ;q,q, q e, q

(7.2.14)

which follows immediately from the fact that (q-n; qh (abqn+l; q) k+l

(1 -

bdqn+l)

(1 - abq2n+l) (1 - bdqk+l)

(1-

+

(q-n;q)k+l (abqn+\q)k af) bdqn+l (1 - abq2n+l) (1 _ bdqk+l) = (q-n, abqn+1; q) k .

The verification that (7.2.11) satisfies the boundary condition PN+l(Xj) = ::::: N is left as an exercise (Ex. 7.3). Note that the 4¢3 series in (7.2.11) remains unchanged if we switch n, a, b, respectively, with j, e, d. This implies that the polynomials Pn(Xj) are self-dual in the sense that they are of degree n in Xj = q-j + edqi+l and of degree j in Yn = q-n + abqn+l and that the weights Wj are obtained from the Vn in (7.2.3) by replacing n, a, b, e, d, by j, e, d, a, b, respectively, i.e.,

o for 0 ::::: j

W· J

=

(edq; q)j (1 - edq2i+l) (eq, aq, bdq; q)j ( )_j abq , (q; q)j(l - edq) (dq, edqla, eqlb; q)j

(7.2.15)

o : : : j ::::: N. Thus {Pn(Xj)};;=o is orthogonal with respect to the weights Wj, while {Pj (Yn)}f=o is orthogonal with respect to the weights Vn (see Ex. 7.4). The calculations have so far been done with the assumption that bdq = q-N, but the same results will hold if we assume that one of aq, eq and bdq is q- N.

180

Applications to orthogonal polynomials

The polynomials Pn(Xj) were discovered by Askey and Wilson [1979] as q-analogues of the Racah polynomials

Pn

F [-n,n+a+ f3 +1,-j,j+1'+6"+1.1]

( ,\.) = 4

J

a

3

+ 1, l' + 1, f3 + 6" + 1

"

(7.2.16)

where ,\j = j(j + l' + 6" + 1), which were named after the physicist Racah, who worked out their orthogonality without apparently being aware of the polynomial orthogonality. In the physicist's language the Pn ('\j) 's are known as Racah coefficients or 6j-symbols. The connection between the 6j-symbols and the 4F3 polynomials (7.2.16) was first made in Wilson's [1978] thesis. The q-analogues in (7.2.11) are called the q-Racah polynomials. We shall now point out some important special cases of the q-Racah polynomials. First, let us set d = b-1q-N-l in (7.2.11) and replace e by be to rewrite the q- Racah polynomials in the more standard notation

Wn(x;a,b,e,N;q)

=

4¢3

[

] q-n,abqn+1, q-X, eqx-N -N ;q,q . aq,q ,beq

(7.2.17)

Then, by (7.1.8), (7.2.3) and (7.2.15), N

LP(x;q)Wm(X;q)Wn(x;q) x=o

=

~m(,n),

(7.2.18)

n q

where

Wn(X; q) p(x; q)

== Wn(x; a, b, e, N; q), == p(x; a, b, e, N; q) (eq-N. q) (1 eq2X-N) (aq beq q-N. q)

=

'x ' , ' x (abq)-X (7.2.19) (q; q)x(l - eq-N)(ea-1q-N, b-1q-N, cq; q)x

and

hn(q)

== hn(a, b, e, N; q) = (bq, aqle; q)N (abq; q)n (1 - abq2n+l) (aq, beq, q-N; q)n (qN ler . (abq2, lie; q)N

(q; q)n(1- abq) (bq, aqle, abqN+2; q)n (7.2.20)

Setting e = 0 in (7.2.17) gives the q-Hahn polynomials

Qn(x)

_

=

. . _ Qn(x, a, b, N, q) -

3¢2

[q-n, abqn+l, q-X . ] aq, q-N ,q, q ,

(7.2.21 )

which were introduced by Hahn [1949a] and, by (7.2.18), satisfy the orthogonality relation N

L Qm(x)Qn(x) (aq; q)x(bq; q)N-x (aq)-X x=o (q; q)x(q; q)N-x 2 (abq2; q)N(aq)-N -----'--(q_;q--'-,)_n('-:-1_-_a_bq--'::),.:-(b.,. . "q,-:-a,-qb_N_+----:;c:-'q)---,n_(_ )n (n)-Nn.r -:N aq q 2 Um n (q; q)N (abq; q)n(1- abq2n+l )(aq, q- ; q)n ' (7.2.22)

7.3 The infinite discrete case

181

for m, n = 0,1, ... , N. Setting a = 0 in (7.2.17) we obtain the dual q-Hahn polynomials (Hahn [1949b])

Rn(P,(x))

== Rn(P,(x); b, e, N; q) _ -

where p,(x)

=

3¢2

[q-n, q-X, eqx-N . ] q-N, beq , q, q ,

(7.2.23)

+ eqx-N, which satisfy the orthogonality relation

q-X

N (eq-N. q) (1 eq2X-N) (beq q-N. q) x ' " Rm(P,(x))Rn(P,(x)) 'x " x qNX-(2)(_beq)-X (q; q)x(1- eq-N)(b-1q-N, eq; q)x

f='o =

(!t

e; ~)N (b (q, b~;;)n) (eq-Nt bm,n, q;q N eq,q ;q n

(7.2.24)

m, n = 0,1, ... , N. For some applications of q-Hahn, dual q-Hahn polynomials, and their limit cases, see Delsarte [1976a,b, 1978], Delsarte and Goethals [1975], Dunkl [1977-1980] and Stanton [1977-1986c].

7.3 The infinite discrete case: the little and big q-Jacobi polynomials As a q-analogue of the Jacobi polynomials (7.1.11), Hahn [1949a] (also see Andrews and Askey [1977]) introduced the polynomials

Pn(x;a,b;q)

= 2¢1(q-n,abqn+\aq;q,xq).

(7.3.1)

It can be easily verified that

. (1-

hm Pn

q--->l

x. a (3. ) _ p~a,(3)(x) - - , q ,q ,q ((3) .

2

Pn a,

(7.3.2)

(1)

He proved that

~ (x. b.) (x. b.) (bq; q)x ( )X _ bm,n ~ Pm q ,a, ,q Pn q ,a, ,q ( .) aq - h ( b. )'

(7.3.3)

< q,aq < 1 and h ( b. ) = (abq; q)n(1- abq2n+l )(aq; q)n(aq; q)= ( )-n

(7.3.4)

x=o

q, q x

n a, ,q

where 0

n a, ,q

(q,. q) n (1 - abq ) (bq,. q)n (abq 2.) ,q =

aq.

Observe that (7.3.3) and (7.3.4) also follow from (7.2.22) when we replace x by N - x and then let N ---+ 00. To prove (7.3.3), assume, as we may, that o :::; m :::; n, and observe that

f

(bq; q)x (aq)xqxkPm(qx; a, b; q) x=o (q; q)x m (q-m abqm+1. q) . ' J qj l¢o(bq;-;q,aq1+k+ j ) j=O (q,aq;q)j

=2:

'

182

Applications to orthogonal polynomials

(7.3.5) by the q-binomial and q-Saalschiitz formulas. Then the orthogonality relation (7.3.3) follows immediately by using (1.5.2) and (7.3.5). It is easy to verify that Pn(x; a, b; q) satisfies the three-term recurrence relation (7.3.6) for n 2: 0, where A

_ (1- abqn+l) (1- aqn+l) _ n n - (1 _ abq2n+l) (1 _ abq2n+2) ( q ),

(7.3.7)

(l-qn)(l-bqn) n C n = (1 _ abq2n) (1 _ abq2n+l) (-aq ),

(7.3.8)

so the condition that AnCn +l > 0 for n = 0,1, ... , is satisfied if 0 < q, aq < 1 and bq < 1. When b < 0 the polynomials Pn(x; a, b; q) give a q-analogue of the Laguerre polynomials L~(1) (x) since liml Pn ((1 - q)x; qOl, _q(3; q)

q-+

= L~OI) (x) / L~OI) (0).

(7.3.9)

Andrews and Askey [1985] introduced a second q-analogue of the Jacobi polynomials, (7.3.10) which has the property that •

•

01

(3

,.

_

hm Pn(x, q ,q ,-q ,q) -

q-+l

p~OI,(3)(X) ((3) , Pn 01, (1)

(7.3.11)

where'Y is real. In view of the third free parameter in (7.3.10) they called the Pn(x; a, b, c; q) the big q-Jacobi and the Pn(x; a, b; q) the little q-Jacobi polynomials. We shall now prove that the big q-Jacobi polynomials satisfy the orthogonality relation

l

aq

cq

( ) ( ) (x/a, x/c; q)oo Pm x; a, b, c; q Pn x; a, b, c; q ( b / . ) dqx x,xc,qoo

(7.3.12)

hn(a, b, c; q)'

where h ( b . ) n a, ,c, q

= M-l(abq;q)n(1_abq2n+l)(aq,cq;q)n(_ ( q; ) q n ( 1 - abq ) (bq, abq /c;) q n

2)-n acq

q

-G) (7.3.13)

7.3 The infinite discrete case

183

and M =

ja

q (x/a, x/c; q)oo dqx cq (x, bx/c; q)oo

aq(1 - q)(q, cia, aq/c, abq2; q)oo (aq,bq,cq, abq/c; q)oo

(7.3.14)

by (2.10.20). Since

j _ja

(x/a, X/C; q)oo (bx/c;q)j(X;q)k ( b / .) dqx cq x, x c, q 00 q (x/a, x/c; q)oo d - cq (k xq, bxql0/ C; q)00 qX aq

_ M(bq,abq/c;q)j(aq,cq;q)k (abq2; q)j+k ' the left side of (7.3.12) becomes M (bq, abq/c; q)m (c/b)m

(aq, cq; q)m m n (q-m abqm+1. q) (q-n abqn+1. q) '"' '"' ' '1 ' ' k j+k x ~ ~ (q,. q)j (q,. q) k (abq 2.) q, j=O k=O ,q j+k 0

(7.3.15)

where we used (7.3.10) and the observation that, by (3.2.2) and (3.2.5),

Pm(x; a, b, c; q) =

(bq,abq/c;q)m( /b)m C ( aq,cq;q ) m

r/,

3'1'2

[q-m,abqm+1,bX/C. ] bq,a bq/ c ' q, q . (7.3.16)

Assume that 0 ::; m ::; n. Since, by (1.5.3)

the double sum in (7.3.15) equals

n-m (qm-n abqm+n+1. q) (q -n abqn+1. q) , ' m ( b m+2) m ' " ' ' , k k (abq2; qhm a q ~ (q, abq2m+2; qh q _

-

(q -n abqn+1. q) ' ' n ( b n+2)n J: (b 2. ) a q Um,n' a q ,q 2n

Substituting this into (7.3.15), we obtain (7.3.12).

(7.3.17)

184

Applications to orthogonal polynomials

7.4 An absolutely continuous measure: the continuous q- ultraspherical polynomials In this and the following section we shall give two important examples of orthogonal polynomials which are orthogonal with respect to an absolutely continuous measure da(x) = w(x)dx. In his work of the 1890's, in which he discovered the now-famous RogersRamanujan identities, Rogers [1893b, 1894, 1895] introduced a set of orthogonal polynomials that are representable in terms of basic hypergeometric series and have the ultraspherical polynomials (7.1.12) as limits when q ---+ 1. Following Askey and Ismail [1983], we shall call these polynomials the continuous q-ultraspherical polynomials and define them by the generating function

((3te iO , (3te- iO ; q) 00 (·0 "0) te' , te-'· q '00

n

00

= "Cn(x; (3lq)t , ~

where x = cos 0, 0 ::; 0 ::; 7f and max(lql, Itl) theorem, it follows from (7.4.1) that

Cn(x; (3lq)

=

(7.4.1)

n=O

< 1. Using the q-binomial

t

((3; qh((3; q)n-k ei(n-2k)O k=O (q; qh(q; q)n-k

-- ((3; q)n inO ( ) e q;q

n

rI-. ( -n (3. 2'1-'1 q "

(3-1 q1-n.,q, q(3-1 e -2iO) .

(742) . .

Note that 1· C ( . AI ) - ~ (A)k(A)n-k i(n-2k)O 1m n x, q q - ~ k'( k)'. e • n k=O = C~(x).

q--->l

(7.4.3)

Before considering the orthogonality relation for Cn(x; (3lq), we shall first derive some important formulas for these polynomials. For 0 < 0 < 7f, 1(31 < 1, set a = q(3-1 e2iO, b = q(3-1, C = qe 2iO and z = (32 qn in (3.3.5) to obtain

2cP1 (q-n, (3; (3-1 q1-n; q, q(3-1 e -2iO) =

" " ((3qn, (3e- 2iO ; q) 00 rI-. (q(3-1 q(3-1 e 2,O. qe 2,o. q (32qn) n+1 -2iO.) 2'1-'1 , " , (q ,e ,qoo ((3, q(3-1, (3q n e2iO , (3-1 q1-n e -2iO; q) 00

+ (qn+1, e 2iO , q(3-1 e -2iO, (3-1 q1-n; q)oo X

2cP1 (q(3-l,q(3-1e-2iO;qe-2iO;q,(32qn).

(7.4.4)

Then, application of the transformation formula (1.4.3) to the two 2cP1 series on the right side of (7.4.4) gives

7.4 An absolutely continuous measure

185

(7.4.5) where

x = cos B.

(7.4.6)

Rewriting the right side of (7.4.5) as a q-integral we obtain the formula

Cn (x· ;3lq) ,

=

2i sin B (;3,;3; q)oo (;32; q) n (l-q)W{3(xlq) (q,;32;q)00 (q;q)n e - iO (queiO,que-iO;q)oo n X e iO (;3ueiO , ;3ue-iO ; q)oo u dqu,

l

(7.4.7)

which was found by Rahman and Verma [1986a]. Now use (1.4.1) to obtain from (7.4.5) that

Cn (x;;3lq) = (;3,;3;;q) ooWi 1(xlq) (;32;q)n (q,;3 ; q)oo (;3q; q)n 2iO inO X {(I - e ) e 2¢1 (q;3-1, qn+\ ;3qn+\ q, ;3e 2iO ) + (1 - e- 2iO ) e- inO 2¢1 (q;3-1, qn+\ ;3qn+l; q, ;3e- 2iO )} 00

= 4sinB Wil(xlq)

L b(k, n;;3) sin(n + 2k + l)B,

(7.4.8)

k=O

where 0

< B < 7[, 1;31 < 1 and b(k,n;;3) = (;3,;3;;q)oo (;32;q)n(q;3-\q)k(q;q)n+k ;3k. (q,;3 ; q)oo (q; q)n(q; q)d;3q; q)n+k

(7.4.9)

The series on the right side of (7.4.8) is absolutely convergent if 1;31 < 1. For Ixl < 1, Iql < 1 and large n it is clear from (7.4.5) that the leading term in the asymptotic expansion of Cn(cosB; ;3lq) is given by

C ( B·;3I) n cos, q

rv

(;3;q)oo {(;3e 2iO ;q)00 -inO + (;3e- 2i O;q)00 ino} (q;q)oo (e 2iO ;q)00 e (e- 2iO ;q)00 e

=2~;3;q?00 q;q

IA(eiO)lcos(nB-a),

(7.4.10)

00

where (7.4.11) and, as elsewhere, f(n) rv g(n) means that lim n--+ oo f(n)jg(n) = 1. For further results on the asymptotics of Cn(x; ;3lq), see Askey and Ismail [1980] and Rahman and Verma [1986a].

186

Applications to orthogonal polynomials

If we use (3.5.4) to express 2¢1 (q-n, (3; (3-1 q1-n; q, q(3-1 e-2iO) as a terminating VWP-balanced S¢7 series in base q~ and then apply (2.5.1), we obtain

,/-. ( -n (3. (3-1 1-n. (3-1 e -2iO) 2'/"1 q " q ,q,q

=

(q ~-n e -2iO.,q) n (_ q(1-n)/2(3-1.,q ~) n ((3 - 1 q21-n ; q21) n

-n/2 ,(31 2 e -iO ,- (31 2 e -iO , -q -n/2 1 1] (3 1_ n /2 -iO 1_ n /2 -iO ; q2, q2 - ,q4 e ,-q4 e (3 n/2 , (312 e iO , (312 e -iO . ~ ~ ] _ ((3 2.'q) n (3-n/2 -inO,/-. [ q-n/2 ,q - ((3 ) e 4 '/"3 1 1 1 1 ,q, q ;q n -(3, (32 q4, -(32 q4 X

4¢3

[q

(7.4.12)

by (2.10.4). However, by (3.10.13),

(7.4.13) and hence, from (7.4.2), (7.4.12) and (7.4.13), we have Cn(cosB; (3lq)

_ ((32;q)n -n/2 [q-n,(32qn,(3~eiO,(3~e-iO. ] - (q,q . ) n (3 4¢3 (3 q2,1 (3 ,- (3 q21 , q, q .

(7.4.14)

Since W,a(cosBlq) = IA(e iO )I- 2 for real (3, it follows from Theorem 40 in Nevai [1979] and the asymptotic formula (7.4.10) that the polynomials C n (cos B; (3lq) are orthogonal on [O,n] with respect to the measure W,a( cos Blq)dB, -1 < (3 < 1. One can also guess the weight function by setting (3 = q>' and comparing the generating function (7.4.1) and the expansion (7.4.8) with the q ---+ 1 limit cases and the weight function (1 - e2iO )>. (1 - e- 2iO )>' for the ultraspherical polynomials C~ (cos B). We shall now give a direct proof of the orthogonality relation

1" where Iql

Cm (cos B; (3lq)Cn (cos B; (3lq)W,a( cos Blq) dB

=

h:(~lq)'

(7.4.15)

< 1,1(31 < 1 and h n ((3lq)

=

(q, (32; ~)oo (q; ;!n(1- (3qn). 2n((3, (3q, q)oo ((3 ,q)n(1- (3)

(7.4.16)

As we shall see in the next section, (7.4.15) can be proved by using (7.4.14) and the Askey-Wilson q- beta integral (6.1.1); but here we shall give a direct proof by using (7.4.2), (1.9.10) and (1.9.11), as in Gasper [1981b]' to evaluate the integral. Since the integrand in (7.4.15) is even in B, it suffices to prove that (7.4.17)

7.4 An absolutely continuous measure

187

when 0 ::; m ::; n. We first show that, for any integer k,

j

7r -7r

e

ike

( I) { 0, if k is odd, W/3 cos () q d() = Ck/2 ((3I)·f . even, q, 1 k IS

(7.4.18)

where

(7.4.19)

I:

By the q-binomial theorem,

=

eike W/3( cos ()Iq)

ff r=Os=O

d()

((3-\ q)r((3-\ q)s (3r+s (q;q)r(q;q)s

r

J-7r

ei(k+2r-2s)e d(),

which equals zero when k is odd and equals

2n

f

s=o

((3-1: q)s((3.-\ q)s+j (31+2s (q, q)s(q, q)s+j

= 2n(~-.\~)j (3j 2¢1 ((3-1, (3-1 qj ; q1+\ q,(32) q, q

when k

=

2j, j

= 0,1, ....

J

By (1.4.5), the above 2¢1 series equals

((3, (3q1+\ q)oo rI-. ( -1 (3-1. (3. (3 j+1) 2\f'1 q , "q, q ((3 2 ,qj+1.) ,q 00 =

((3,(3,;!q)oo (q;.q)j (1+qj). (q,(3 ,q)oo ((3q,q)j

From this and the fact that W /3 ( cos () q) is symmetric in (), so that we can handle negative k's, we get (7.4.18). Hence, from (7.4.2), 1

(7.4.20) equals zero when n - k is odd and equals

(7.4.21 ) when n - k = 2j is even. From (1.9.11) it follows that this 4¢3 series and hence the integral (7.4.20) are equal to zero when n > Ikl. Hence (7.4.17) holds when m -I- n. If k = ±n -I- 0, then (1.9.10) gives

I:

eikOCn(cos();(3lq)W/3(cos()lq) d()

2n((3, (3q; q)00((32; q)n (q, (32; q)oo((3q; q)n

(7.4.22)

188

Applications to orthogonal polynomials

from which it follows that (7.4.17) also holds when m

=

n.

7.5 The Askey-Wilson polynomials In view of the 4¢3 series representation (7.4.14) for the continuous q-ultraspherical polynomials it is natural to consider the more general polynomials

(7.5.1) which are of degree n in x = cosO, and to try to determine the values of cv, /3, ,,(, 8, E for which these polynomials are orthogonal. Because terminating balanced 4¢3 series can be transformed to other balanced 4¢3 series and to VWP-balanced 8¢7 series which satisfy three-term transformation formulas (see, e.g., (7.2.13), (2.11.1), Exercise 2.15 and the three-term recurrence relation for the q-Racah polynomials), one is led to consider balanced 4¢3 series. From Sears' transformation formula (2.10.4) it follows that if we set cv = abcdq-l, /3 = a, "( = ab, 8 = ac and E = ad, then the polynomials

=

(ab, ac, ad; q)na-n

4¢3

[

q-n abcdqn-l ae iIJ ae- iIJ ] ' b 'd' ; q, q a ,ac,a

(7.5.2) are symmetric in a, b, c, d. In addition, for real 0 these polynomials are analytic functions of a, b, c, d and are, in view of the coefficient (ab, ac, ad; q)na-n, realvalued when a, b, c, d are real or, if complex, occur in conjugate pairs. Askey and Wilson [1985] introduced these polynomials as q-analogues of the 4F3 polynomials of Wilson [1978, 1980]. Since they derived the orthogonality relation, three-term recurrence relation, difference equation and other properties of Pn(x; a, b, c, dlq), these polynomials are now called the AskeyWilson polynomials. Since the three-term recurrence relation (7.2.1) for the q-Racah polynomials continues to hold without the restriction bdq = q-N, by translating it into the notation for Pn(x; a, b, c, dlq) we find, as in Askey and Wilson [1985], that the recurrence relation for these polynomials can be written in the form 2xPn(x) = AnPn+l(X)

+ BnPn(x) + CnPn-l(X),

n 2 0,

(7.5.3)

with P-l(X) = O,Po(x) = 1, where An =

1 - abcdqn-l

~------~~~------~~

(1 - abcdq2n-l) (1 - abcdq2n) ' (1 - qn)(1 - abqn-l )(1 - acqn-l )(1 - adqn-l) (1 - abcdq2n-2) (1 - abcdq2n-l) X (1- bcqn-l)(I- bdqn-l)(I- cdqn-l),

(7.5.4)

(7.5.5)

7.4 An absolutely continuous measure

189

and

= a + a-I

Bn

-

A n a- l (l- abqn)(l - aeqn)(l - adqn)

- Gna/ (1 - abqn-I) (1 - aeqn-I) (1 - adqn-I).

(7.5.6)

It is clear that An, B n , G n are real if a, b, e, d are real or, if complex, occur in conjugate pairs. Also AnGn+1 > 0, n = 0,1, ... , if the pairwise products of a, b, e, d are less than 1 in absolute value. So by Favard's theorem, there exists a measure da(x) with respect to which Pn(x; a, b, e, dlq) are orthogonal. In order to determine this measure let us assume that max(lal, Ibl, lei, Idl, Iql) < 1. Then, by (2.5.1), Pn( cos B; a, b, e, dlq) (ab, ae, be, de- iO , ; q)

=

"0

netn

(abee iO ; q)n S W7 (abee iO q-\ ae iO , be iO , ee iO , abedqn-I, q-n; q, qd-Ie- iO ) , (7.5.7)

X

and, by (2.11.1), S

W (abeeiOq-l. ae iO beiO ee iO abedqn-I q-n. qd-Ie- iO ) 7

""

"

q) 0 ( abee iO , bedeiOqn , beiOqn+1 , eeiOqn+1 , ae- iO , be- iO , ee- iO , de-iOqn.' 0 2iO 2iO ab ae be qn+1 bdqn edqn beqn+Ie e. q) ( ,

X S W7

"

,

"

"

CXJ

(beq n e 2iO ; beqn, beiO , ee iO , qa-Ie iO , qd-Ie iO ; q, adqn)

(abeeiO, abee- iO qn, bede- iO qn, be- iO qn+l, ee- iO qn+l; q) 00 +~--~~~~--~--~~~~--~~~~~--~

X

(qn+l, bdqn, edqn, abeeiOqn, beqn+Ie- 2iO ; q)oo q) 0 -2tnO " ( ae iO , beiO , ee iO , de iO '. 0 ( ab, ae, ad, e2"0) e t ; q 00

X SW 7

where

(beqn e -2iO; beqn, be- iO , ee- iO , qa-Ie- iO , qd-Ie- iO ; q, adqn) , (7.5.8)

°< B
3

[

q-n, _q-n, _q~-n, q~-A-n. ] ql-A-2n, q~ -n e 2iO, q~ -n e -2iO , q, q .

(7.5.39)

series, we obtain

C2n (cos e; qAlq) _ -

(q A qA+~. q) ' ' n -n/2 ( 1 ) q q, q2; q

1

4¢>3

.

1

.

[q-n,qn+A,q2e2.0,q2e-2.0. ] A+~ _ ~ _ ' q, q . q ,q, q

n

(7.5.40) This, together with (7.5.25), yields (7.5.35). The proof of (7.5.36) is left as an exercise. Following Askey and Wilson [1985] we shall obtain what are now called the continuous q-Hahn polynomials. First note that the orthogonality relation (7.5.15) can be written in the form

1:

Pm(cose; a, b, c, dlq)Pn(cose; a, b, c, dlq)w(cose; a, b, c, dlq) sine de

hn(a, b, c, dlq)'

(7.5.41)

194

Applications to orthogonal polynomials

Replace B by B + ¢, a by ae i¢, b by ae- i¢ and then set c = be i¢, d = be- i ¢ to find by periodicity that if -1 < a, b < 1 or if b = a and lal < 1, then

[ : Pm(cos(B + ¢); a, blq)Pn(cos(B + ¢); a, blq)W(B) dB (7.5.42) where

(7.5.43) (7.5.44) and

x (1 _ a2b2q-l) (q , a2 , b2 , ab " ab abe 2i ¢ , abe- 2i ¢.'q) n

(7.5.45) The recurrence relation for these polynomials is 2 cos(B + ¢ )Pn(cos(B + ¢); a, blq)

= AnPn+l (cos(B + ¢); a, blq) + BnPn(cos(B + ¢); a, blq)

+ CnPn-l(cos(B + ¢); a, blq)

(7.5.46)

for n = 0,1, ... , where P-l (x; a, blq) = 0,

A -

1 - a 2 b2 qn-l

(7.5.4 7)

-,---..".-:-;:,......."..----:-..,...-7-----,,-:-:::-;:--:-

n - (1 _ a 2b2q2n-l) (1 _ a 2b2q2n) '

and

Bn

=

ae i¢ + a-1e- i¢ - Ana-1e-i¢(1 - a 2qn)(1 - abqn)(l - abe 2i ¢qn) - Cnae i¢ /(1- a 2qn-l)(1_ abe 2i ¢qn-l)(1_ abqn-l). (7.5.49)

If we set a obtain

= qO

1 we

(7.5.50)

7.6 Connection coefficients

195

where . (3) _ · n Pn ( x, a, - 2

D

3£2

[-n, n+ 2aa++(32(3, 2-a 1, a- ix ,.1].

(7.5.51)

For further details and an extension to one more parameter, see Askey [1985b], Askey and Wilson [1982, 1985], Atakishiyev and Suslov [1985], and Suslov [1982, 1987].

7.6 Connection coefficients Suppose fn(x) and gn(x), n = 0,1, ... , are polynomials of exact degree n in x. Sometimes it is of interest to express one of these sequences as a linear combination of the polynomials in the other sequence, say, n

gn(X)

=

L ck,nfk(X).

(7.6.1)

k=O

The numbers Ck,n are called the connection coefficients. If the polynomials fn(x) happen to be orthogonal on an interval I with respect to a measure da(x), then Ck,n is the k-th Fourier coefficient of gn(x) with respect to the orthogonal polynomials fk(X) and hence can be expressed as a multiple of the integral II fk(X)gn(x)da(x). A particularly interesting problem is to determine the conditions under which the connection coefficients are nonnegative for particular systems of orthogonal polynomials. Formula (7.6.1) is sometimes called a projection formula when all of the coefficients are nonnegative. See the applications to positive definite functions, isometric embeddings of metric spaces, and inequalities in Askey [1970, 1975], Askey and Gasper [1971]' Gangolli [1967] and Gasper [1975a]. As an illustration we shall consider the coefficients Ck,n in the relation n

Pn(x; a, (3, 1', dlq) =

L Ck,nPk(X; a, b, c, dlq)·

(7.6.2)

k=O

Askey and Wilson [1985] showed that

Ck,n

=

(ad,(3d,1'd,q;q)n (a(31'd qn-\q)k k2-nkdk-n kq (ad, (3d, 'I'd, q, abcdq 1; qh (q; q)n-k

x S¢4

[qk-n, a(31'dqn+k-1, adqk, bdqk, cdqk . ] abcdq2k, adqk, (3dqk, 1'dqk ,q, q .

To prove (7.6.3), temporarily assume that max observe that, by orthogonality,

bk,j =

11

w(x;a,b,c,dlq)Pk(x;a,b,c,dlq)(deiO,de-iO;q)j dx

-1

vanishes if j


0 when 0 < a < a < 1, 0 < q < 1 and max (be, bd, cd, abed) < 1. Another simple case is when the 5rP4 series in (7.6.3) reduces to a summable 4rP3 series. For example, this happens when we set d = q!, c = -q! = 'Y, b = -a and (3 = -a. The 5rP4 series then reduces to

(7.6.10) which equals ( a2qn+k+1 qk-n+1 a2qk+n+2 a2qk-n+2/a2.q2) '

,

,

(q,a2q2k+1,a2q2k+2,a2q2/a2;q2)00

,

00

(

()/ n+k) n-k 2 a q 2

(

)

7.6.11

by Andrews' formula (Ex. 2.8). Clearly, this vanishes when n - k is an odd integer and equals

(qa 2, q2a 2; q2)n_2 o(q, 00 2 /a 2; q2)Jo (qa 2, q2a 2; q2)n_j

J -----;-----;c----';;:~'____;;co__------"(a 2q2n+ 1-4 j )j

(7.6.12)

7.7 A difference equation and a Rodrigues-type formula when n - k

197

2j,j = 0, 1, ... , [~]. Since by (3.10.12) and (7.4.14)

=

.

Pn (X, a,

__ ~ a,

~I) _

(q2,qa 2 ;q2)n (. 21 2) (2.) On X, a q , a ,q n

q ,q q -

(7.6.13)

we obtain, after some simplification, Rogers' formula

o (. n

X, "(

1) q

= [~l ~

k=O

(3k("( j (3; qh(,,(; q)n-k(l - (3qn-2k) 0 (. (31 ) ( . ) ((3 .) (1 _ (3) n-2k X, q q, q k q, q n-k

(7.6.14)

after replacing 0;2, a 2 , q2 by ,,(, (3 and q, respectively. It is left as an exercise (Ex. 7.15) to show that formulas (7.6.2) and (7.6.14) are special cases of the q-analogue ofthe Fields-Wimp formula given in (3.7.9). For other applications of the connection coefficient formula (7.6.2), see Askey and Wilson [1985].

7.7 A difference equation and a Rodrigues-type formula for the Askey-Wilson polynomials

The polynomials Pn(x; a, b, c, dlq), unlike the Jacobi polynomials, do not satisfy a differential equation; but, as Askey and Wilson [1985] showed, they satisfy a second-order difference equation. Define Eif(e iO )

6qf (e iO )

= f (q±~eiO), (Et - E;;) f (e iO )

=

Dqf(x) = 6q!(X) , uqx

and

(7.7.1)

X = cose.

(7.7.2) (7.7.3)

Clearly,

~ (q~

6q( cos e) =

and 6q (ae iO , ae- iO ; q) n

q-~)

-

(e iO - e- iO ) = -iq- ~ (1 - q) sin e,

= 2aiq- ~ (1 -

qn) sin

e (aq~ eiO , aq~ e- iO ; q)n-l,

(7.7.4) (7.7.5)

so that "0

D q ( ae'

"0)

,

ae-'·'qn

=-

2a(1-qn)

(1 _ q)

1"0

1

"0

(aq2 e' , aq2 e-' ,. q) n-l,

(7.7.6)

which implies that the divided difference operator Dq plays the same role for (aeiO,ae-iO;q)n as djdx does for xn. When q ---+ 1, formula (7.7.6) becomes

d~ (1- 2ax + a 2)n = -2an(1- 2ax + a 2t- 1 . Generally, for a differentiable function .

hm Dqf(x)

q--+l

f (x) =

we have

d

-d f(x). X

198

Applications to orthogonal polynomials

Following Askey and Wilson [1985], we shall now use the operator Dq and the recurrence relation (7.5.3) to derive a Rodrigues-type formula for Pn(x; a, b, c, dlq). First note that by (7.5.2) and (7.7.5)

8q Pn(x; a, b, c, dlq) 1 1 1 1 1 n sine (1 - qn)(l - qn- 1 abcd)Pn-l(X; aq2, = -2iq2bq2, cq2, dq2Iq). (7.7.7) If we define

(7.7.8) and

r (eiO ) = n

4'" 'f'3

[q-n,abCdqn-l,aeiO,ae-iO. ] ab , ac , ad ' q, q ,

(7.7.9)

then the recurrence relation (7.5.3) can be written as

q-n(1- qn)(l_ abcdqn-l)rn(e iO ) = A( -e) [rn(q-1e iO ) - rn(e iO )] + A(e) [rn(qe iO ) - rn(e iO )].

(7.7.10)

Also, setting (7.7.11) (7.7.12) we find that (7.7.10) can be expressed in the form

q-n(1- qn)(l_ abcdqn-l)V(eiO)V(e-iO)Pn(x) = 8q [{E:V(e iO )} {E';-V(e- iO )} {8 qPn(xH]·

(7.7.13)

Combining (7.7.7) and (7.7.13) we have

- q-n/2V(e iO)V(e- iO)Pn(X; a, b, c, dlq) = 8q [{ E:V(e iO )} {E.;-V(e- iO )} (e iO _ e- iO ) x Pn-l (X; aq! , bq! , cq! , dq! Iq)] . (7.7.14) Since

(e iO _ e- iO ) E:V(eiO)E';-V(e- iO ) (e iO -

c iO ) (qe 2iO , qe- 2iO ; q)

00

h( cos 61; aq! )h( cos 61; bq! )h( cos 61; cq! )h( cos 61; dq!) W (eiO;aq!,bq!,cq!,dq!lq) eiO

-

e- iO

199

Exercises (7.7.14) can be written in a slightly better form

q-n/2W(eiO ; a, b, c, dlq)Pn(x; a, b, c, dlq)

= liq [(e iO

- e-iO)-l W(e iO ; aq~, 1

1

1

1

bq~, cq~ ,dq~ Iq) ]

Pn-l (x; aq2 ,bq2 ,cq2, dq2lq) .

X

(7.7.15) Observing that, by (6.3.1),

Vl- x 2w(x; a, b, c, dlq) =

W(e iO ; a, b, c, dlq)

we find by iterating (7.7.15) that

w(x; a, b, c, dlq)Pn(x; a, b, c, dlq) =

(_I)k

(1 ;

q) k qnk/2-k(k+l)/4

k k k k k k k k ] x Dqk[ w(x; aq2, bq2, cq2, dq2Iq)Pn_k(X; aq'i, bq'i, cq'i, dq'ilq)

=

(-It

(1 ;

q) n q(n 2 -n)/4 D~ [w(x; aqn/2, bqn/2, cqn/2, dqn/2Iq)] . (7.7.16)

This gives a Rodrigues-type formula for the Askey-Wilson polynomials. By combining (7.7.7) and (7.7.15) it can be easily seen that the polynomials Pn(x) = Pn(x; a, b, c, dlq) satisfy the second-order difference equation 1

1

1

1

Dq [ w(x; aq2, bq2, cq2, dq 2 Iq)D qPn(x)

]

+ AnW(X; a, b, c, dlq)Pn(x) = 0,

(7.7.17)

where (7.7.18)

Exercises 7.1 If {Pn(x)} is an orthogonal system of polynomials on (-00, (0) with respect to a positive measure da(x) that has infinitely many points of support, prove that they satisfy a three-term recurrence relation of the form

XPn(X)

=

AnPn+l(X)

+ BnPn(x) + CnPn-l(X), n;::: 0,

with p-l(X) = 0, po(x) = 1, where An, B n , Cn are real and AnCn+1 > for n ;::: 0. 7.2 Let PO(X),Pl(X), ... ,PN(X) be a system of polynomials that satisfies a three-term recurrence relation

°

200

Applications to orthogonal polynomials n = 0,1, ... , N, where Darboux formula

P-l (x)

= O,Po(x) = 1. Prove the Christoffel-

n

(x - Y) LPj(x)pj(Y)Vj = Anvn [Pn+1 (x)Pn (y) - Pn(X)Pn+l(Y)] ' j=O :S n :S N, where Vo = 1 and vnCn = vn-1An- 1, 1 :S n :S N. Deduce

°

that

n

LP](x)Vj = Anvn [p~+1(X)Pn(X) - P~(X)Pn+l(X)l j=O and hence

N

LP](Xk)Vj j=O

=

ANVNPN(Xk)p~+l(Xk)

if x k is a zero of PN +1 (x) .

= N, the recurrence relation (7.2.1) reduces to (1 - q-j) (1 - cdqHl) PN(Xj) = CN [PN(Xj) - PN-l (Xj)] ,

7.3 Show that when n

where Pn(Xj) is given by (7.2.11), Xj by (7.2.9), and An and Cn by (7.2.5) and (7.2.6). Hence show that (7.2.1) holds with x = Xj,j = O,l, ... ,N. (Askey and Wilson [1979]) 7.4 If Pn(Xj), Vn and Wj are defined by (7.2.11), (7.2.3) and (7.2.15), respectively, prove directly (i.e. without the use of Favard's theorem) that N

N

LPm(Xj)Pn(Xj)Wj = j=O and

V;;:-l

N

L Wj Dm,n j=O N

LPn(Xj)Pn(Xk)Vn = wjl L Wn Dj,k. n=O n=O [Hint: First transform one of the polynomials, say Pn(x), to be a multiple of the

4¢3

series on the left side of (7.2.14)].

7.5 Let one of a, b, c, d be a nonnegative integer power of q-l and let

¢(a, b) =

4¢3 [

a~~~/~d; q, q] ,

where efg = abcdq. Prove the Askey and Wilson [1979] contiguous relation

A¢(aq-l, bq) where

+ B¢(a, b) + C¢(aq, bq-l) = 0,

A = b(l - b)(aq - b)(a - e)(a - f)(a - g), B = ab(a - bq)(b - a)(aq - b)(l - c)(l - d) - b(l - b)(aq - b)(a - e)(a - f)(a - g) + a(l - a)(a - bq)(e - b)(f - b)(g - b), C = -a(l - a)(a - bq)(e - b)(f - b)(g - b),

Exercises

201

and derive the following limit case which was found independently by Lassalle [1999]: ( (c-b)+ ab(1-c) _ aC(l-b)) y x _ (l-c)(y-a)(y-b) y(l - y) -

3

(l-b)(x-a)(x-c) x(1- x)

3(P2

(a,b,C;q, x y ) x,y abc

. +"2) (1) Ln/kJ (>')rn/kl

hn =

where the roof and floor functions are defined by

Ial = laJ =

smallest integer greater than or equal to a, largest integer less than or equal to a.

(AI-Salam, Allaway and Askey [1984b]) 7.25 The sieved ultraspherical polynomials of the second kind are defined by B~(x; k)

= lim Cn(x; sAk+1WklsWk), Wk = exp(27rijk). 8---+1

Show that B~(x; k) satisfies the recurrence relation 2xB~(x; k)

=

B~+l (x; k)

+ B~_l (x; k),

n

+ 1 -=I- mk,

2x(m + >')B~k-1 (x; k) = mB~k(x; k) + (m + 2>')B~k_2(X; k), where B6(x; k) = 1; Bf(x; k) = 2x if k 2:: 2; Bf(x; 1) = 2(>' + l)x. Show also that B~(x; k) satisfies the orthogonality relation

1 B~(x; k)B~(x; 1

k)w(x) dx =

-1

8~,n, n

where

k-1 W(X) = 22A (k-1)(1_ x 2)!

II Ix

2 -

cos 2(7rjjk)IA

j=O

and

_ hn

-

(l)Ln/kJ(>'+l)L~J

2r(>'+1) 1

1

(

nt J

r("2)r(>. +"2) >. + 1) Ln/kJ ( 2>' + 1) L

(AI-Salam, Allaway and Askey [1984b])

1

.

206

Applications to orthogonal polynomials

7.26 Using (2.5.1) show that

_ (ab,ae,be,q;q)n ( . b,e, dl) PnX,a, q - (bd-l.) aeq ,qn n (abeq-1 eiO;q)k (1- abee iO q2k-1) (aeiO,beiO,eeiO;q)k x {; (q; qh(l - abeq- 1eiO ) (be, ae, ab; q)k x (abedq-1; q)nH (de- iO ; q)n-k ei(n-2k)O

(abee 20 ; q)nH (q; q)n-k Deduce that the polynomials

Pn(x)

= q-+O lim Pn(x;a,b,e,dlq)

are given by

PO(X) = 1 = Uo(x), P1(X) = (1- 84)U1(X)

+ (83 - 81)UO(X), P2(X) = U2(x) - 81U1(X) + (82 - 84)UO(X), 4

Pn(X) = ~) -1)k8k Un _k(X), k=O

n ~ 3,

where 80

= 1,

81

= a + b + e + d, 83

82

= ab + ae + ad + be + bd + cd,

= abc + abd + aed + bcd,

84

= abed,

and Un (COS 0) = sin(n+1)ej sinO, U- 1(x) = O. When max(lal, Ibl, lei, Idl) < 1 show that these polynomials satisfy the orthogonality relation

211 7r

-1

Pm(X)Pn(x)(l - x 2)! dx (1- 2ax + a 2 )(1 - 2bx + b2 )(1 - 2ex + e2 )(1 - 2dx + d 2 )

0,

1- abed = { (1 - ab)(l - ae)(l - ad)(l - be)(l - bd)(l _ cd)' 1- abed, 1,

m#n, m m m

= n = 0, = n = 1, = n ~ 2.

(Askey and Wilson [1985]) 7.27 Prove that

· _ q, q! , _! _ ( n+2.,q) n sin(n. +0 1)e ' q Iq) - q S1n

(i)

Pn (cos 0, q,

(ii)

Pn(cos 0; 1, -1, q!, -q! Iq) = 2(qn; q)n cos nO,

(iii)

Pn(cosO,q,

(iv)

Pn(cosO,l,

n ~ 1,

· _ 1,q, ! _! _ ( n+ 1.,q) n sin( n + ~)e q Iq) - q sin(Oj2) , · _

! _!

_ ( n+1. ) cos(n + ~)e

q,q , q Iq) -

q

,q n

cos(Oj2) .

Exercises

207

7.28 Use the orthogonality relations (7.5.28) and (7.5.29) to prove the quadratic transformation formula (7.5.26). 7.29 Verify the orthogonality relations (7.5.28) and (7.5.29). 7.30 Verify formula (7.5.36). 7.31 Suppose that a,b,c,d are complex parameters with max (Ibl, Icl, Idl, Iql) < 1 < lal such that laqN+11 < 1 < laqNI, where N is a nonnegative integer. Use (6.6.12) to prove that 1

[1 Pm(X)Pn(X)W(X; a, b,

N

c, dlq) dx

+ 2n ~Pm(Xk)Pn(Xk)Wk

hn(a, b, c, dlq)' where Pn(x) by (7.5.22).

=

Pn(x;a,b,c,dlq), Xk

=

~

(aqk +a- 1 q-k) and Wk is given

7.32 Prove that

(i)

p~a,(3)(_x;q)

(ii)

p~a,(3)(_xlq)

= (-ltP~f3,a)(x;q), = (_ltq(a-f3)n/2p~f3,a)(xlq).

7.33 Using (7.4.1), (7.4.7) and (2.11.2) prove that

~ (~;;qJ)n On(x; ,8lq)On(Y; ,8lq)tn

(i)

(t 2,,8;q)= (,8t2, ,82; q)= x 8 W7

(~t2q-l. ~ jJ ,jJ,

teiiJ+i¢ , te -iO-i¢ , te iO - i¢ , tei¢-iO., q,jJ~)

,

~ (q; q)n 1 - ,8qn C ( ~I)O ( ~I) n f=' o (,82;q)n 1-,8 n X;{J q n Y;{J q t

(ii)

(t2,,8; q) = (,8teiiJ+ i¢, q,8te iO - i¢; q) = (q,8t2, ,82; q)= (teiiJ+i¢, te iO - i¢; q)= x

SW7

2

(,8t 2;,8, qteiiJ+ i¢, qte- iO - i¢, te iO - i¢, tei¢-iO; q,,8) ,

where -1 < q,,8, t < 1 and x = cosB, y = cos¢. (Gasper and Rahman [1983a], Rahman and Verma [1986a]) 7.34 Show that

Pn(cosB; a, b, c, dlq) x

l

qe - W / d

qeW

/d

D-l(B)(ab, ac, bc; q)n (dueiO,due- iO , abcdujq; q) =

(qju;q) (dU)n n du (daujq, dbujq, dcujq; q)= (abcdujq; q)n q q,

=

where

D( B) =

-iq(l - q) 2d (q, ab, ac, bc; q)=h( cos B; d)w( cos B; a, b, c, dlq).

208

Applications to orthogonal polynomials Hence show that 00 ( 11 ) n '""" a2c2) ; q nPn (() cos ; a, aq 1 2 , C, cq 2 q ~ 1 1 t n=O (q; q)n(a 2q2, ac, acq2; q)n

(at " ac2t -acteiIJ , -acte- iIJ ., q) 00 (-ct, -a 2ct, te iIJ , te- iIJ ; q)oo

x 8 W 7 ( -a 2ctq-\ -ac, -a/c, -ctq-~, ae iIJ , ae- iIJ ; q, -ctq~) . (Gasper and Rahman [1986]) 7.35 Show that Ex. 6.9 is equivalent to

1 1

w(y; a, b, p,eiIJ, J.Le-iIJlq)Pn(Y; a, b, c, dlq) dy

-1

27r(abJ.L 2; q)oo (ab; q)n n (q, ab, J.L2, aJ.Le iIJ , aJ.Le- iIJ , bJ.LeiIJ , bJ.Le- iIJ ; q)oo (abJ.L2; q)n J.L x Pn(cos (); aJ.L, bJ.L, CJ.L- 1 , dJ.L- 1 Iq),

where max(lal, Ibl, IJ.LI) 7.36 Show that if for Iql


-+I, qn+v+~ ,qn+2v, qn+v+leilJ, qn+v+le-ilJ . ] q2n+2v+l, qn+v+>-+I, qn+2v+l, _qn+v+>-+I, _qn+v+~ ,q, q

Show that Un(x) satisfies the q-differential equation

Dq[(qn+V+ 1 e iIJ , qn+v+l e -ilJ; q)-nUn(x)]

=X

1 - qn+2v 1 - qn+l

3

.

3.

(qn+v+2 e,IJ, qn+v+2 e-,IJ; q)

-2n-2v-l

Dq[(q-n-v e iIJ , q-n-v e- iIJ ; q)n+2v+l Un+1 (x)].

(Gasper [1989b])

Exercises

209

7.38 Show that the discrete q-Hermite polynomials

[n/2] (q; q)n (_l)k k(k-l) n-2k H ( . )= " n X, q ~ (2 2) ( ) q X k=O q; q k q; q n-2k satisfy the recurrence relation

and the orthogonality relation

where 1jJ(x) is a step function with jumps

Ixl (X 2q2, q; q2)= 2

(q2; q2)=

at the points x = ±qj, j = 0,1,2, .... (Ai-Salam and Garlitz [1965], Ai-Salam and Ismail [1988]) 7.39 Let a

< 0 and 0 < q < 1. Show that

I:

u$::)(x;q)u~a)(x;q) doJa) (x)

= (1- a)( _a)n(q; q) nq G)8m ,n, where

and oJa) (x) is a step function with jumps

qk

at the points x

=

qk, k

=

0,1, ... , and jumps

-aqk

at the points x = aqk, k = 0,1,.... Verify that when a = -1 this orthogonality relation reduces to the orthogonality relation for the discrete q-Hermite polynomials in Ex. 7.38. (See Ai-Salam and Garlitz [1957, 1965], Ghihara [1978, (10.7)], and Ismail [1985b, p.590]) 7.40 Show that if

(q; q)n k h ( . )- ~ n x, q - ~ ( ) ( ) x, k=O q; q k q; q n-k

210

Applications to orthogonal polynomials then h~(x;q) - hn+l(x;q)hn-1(x;q)

n

qn-kxk

k=O

q, q n-k

= (1 - q)(q; q)n-l L h~_k(X; q) ( .)

, n ;::: 1.

Deduce that the polynomials hn(x; q), which are called the Rogers-Szego polynomials, satisfy the Thnin-type inequality

for x ;::: 0 when 0 (Carlitz [1957b])

< q < 1 and n = 1,2, ....

7.41 Derive the addition formula

Pn(qZ; 1, 1; q) py(qZ; qX, 0; q) =

Pn(qx+y; 1, 1; q) Pn(qY; 1, 1; q) py(qZ; qX, 0; q)

+ Ln k=l X

(q. q) (q. q) qk(k+y-n) ' x+y+k , n+k (q; q)x+y(q; q)n-k(q; q)~

Pn_k(qx+y; l, qk; q) Pn-k(qY; l, l; q)

Z x x Py+k(q ; q ,0; q) X

n (q; q)y(q; q)n+kqk(x+y-n+l)

+ k=l L (.) (.) (. )2 q,qy-kq,qn-kq,qk

Pn_k(qx+y-k; l, l; q) Pn-k(qy-\ qk, qk; q) Py_k(qZ; qX, 0; q)

where x, y, z, n = 0,1, ... , and Pn(t; a, b; q) is the little q-Jacobi polynomial defined in Ex. 1.32. (Koornwinder [1991a]) 7.42 Derive the product formula 00

Pn(qX; 1, 1; q) Pn(qy; 1, 1; q)

=

(1 - q) LPn(qZ; 1, 1; q) K(qX, qy, qZ; q)qZ, z=O

where x, y, z, n = 0,1, ... , Pn(t; a, b; q) is the little q-Jacobi polynomial, and x+l y+l z+l. ) K( x y z. ) _ ( q ,q ,q ,q 00 xy+Xz+yz q ,q ,q ,q (1 -)( q q, q,.) q 00 q

x {3¢2(q-X,q-y,q-Z;0,0;q,q)}2. (Koornwinder [1991a]) 7.43 The q-Laguerre polynomials are defined by

Exercises

211

> -1 then these polynomials satisfy the orthogonality

Show that if a relation

(i)

and the discrete orthogonality relation (ii) where

A

=

(q, -c(1 - q)qe3

219

polynomials

Then, replacing j by j - k in the sum on the right side of (8.2.4), we obtain 41>3 [

q-n, aqn, bl , b2 ] j ;q,q b, b3 , qab l b2 bb3

~

[ q-n, aqn,

CI, C2

j

41>3

]

C3, qacIC2 CC3

C,

;q,q

(q-n, aqn, CI, C2, qabdbb3, qab 2 jbb 3 ; q)m m (q, C, C3, qajbb 3 , qacIC2jcC3, qab l b2 jbb 3 ; q)m q

=~

~ (bb 3q-m-1 ja; q)j(l - bb3q 2j-m-1 ja)(b l , b2 , bq-m ja, b3q-m ja, q-m; q)j

X

~

(q; q)j(l - bb3q-m-1 ja)(bb 3q-m jab l , bb3q-m jab 2 , b3 , b, bbJ/a; q)j

bb3qm )j

x ( --

[q-j, ql-m jc, ql-m j C3, bb3qj-m-1 ja, CC3q-m j acIC2. 51>4

bl b2

q

I

-m JCI,

q

I

-m j C2,

'

bq-m ja, b3q-m ja

]

q, q . (8.2.5)

Note that the 51>4 series in (8.2.5) is balanced and, in the special case C = aqjb and C3 = aqjb3 , becomes a 31>2 which is summable by (1.7.2). Thus, we obtain the formula

bb3c2jaq, q

where). formula

=

I -m

bq-m ja,

j C2,

b3q-m ja,

b3 ,

q-m

b,

bb3 ja,

;

q,

a q2 bl b2CIC2

]

,(8.2.6)

bb3q-m-1 ja. This formula is a q-analogue of Bailey's [1933] product 2FI ( -n,

a

+ n; b; x)

2FI ( -n,

a

+ n; 1 + a -

b; y)

= F 4 ( -n, a + n; b, 1 + a - b; x(l - y), y(l - x)),

(8.2.7)

~ ~ (a)m+n(b)m+n m n F ( b d ) 4 a, ;C, ;x,y = ~~ m!n!(c)m(d)n x y .

(8.2.8)

where

However, even though (8.2.6) is valid only when the series on both sides terminate, (8.2.7) holds whether or not n is a nonnegative integer, subject to the absolute convergence of the two 2FI series on the left and the F4 series on the right. Application of Sears' transformation formula (2.10.4) enables us to transform one or both of the 41>3 series on the left side of (8.2.6) and derive a number of equivalent forms. Two particularly interesting ones are

220

Further applications

(8.2.9)

bb3 C2/ aq, ql-m /C2,

b3 ,

(8.2.10)

b,

where ,\ = bb3 q-m-l/a. Either of the formulas (8.2.9) and (8.2.10) may be regarded as a q-analogue of Watson's [1922] product formula for the Jacobi polynomials 2FI( -n, a + n; b; x) 2FI( -n, a + n; b; y)

(-1 )n(1+a-b)n (b)n F4 (-n,a+n;b,l+a-b;xy, ( I-x )( l-y )) ,

=

(8.2.11) where n = 0,1, .... The special case in which the lOcP9 series in (8.2.6), (8.2.9) or (8.2.10) become balanced is also of interest in some applications. Thus, if we set C2 = aq/bIb2CI, then by using Bailey's transformation formula (2.10.8) we may express (8.2.9) in the form 4cP3

[

q-n, aqn, bl , b2 ] b, b3, qa bb/bb ;q,q I 2 3

= (aq/b,aq/b 3 ;q)n (bb 3 )n (b, b3 ; q)n X lOcP9

[

fJ'

aq

4cP3

t

] [q-n, aqn, bb3 cd aq, bb3 /bl b2CI b, b3, bb/bb ;q,q 3 I 2

(q-n,aqn,bICI,b2CI,aq/bIb2CI;q)mqm m=O (q, aq/b, aq/b 3 , bIb2CI, bb3 /b Ib2; q)m bIb2CI/b,

bIb2CI/b 3 ,

b,

b3 ,

8.3 Product formulas for q-Racah and Askey-Wilson polynomials

bb3cI/aq,

aqm,

qab 1b2/bb 3,

b1b2c1q-m la,

221

(8.2.12)

where /-l = b1b2c1q-1. This provides a q-analogue of Bateman's [1932, p. 392] product formula 2F1 ( -n, a + n; b; x) 2F1 ( -n, a + n; b; y)

= (_I)n (1 + a (b)n

b)n:t (-n)k(a + n)k (1- x _ y)k k=O k!(1 + a - bh

x 2F1 (-k, a + k; b; -xy/(1 - x - y)).

(8.2.13)

8.3 Product formulas for q-Racah and Askey-Wilson polynomials Let us replace the parameters a,b,b 1,b2,b3,C1,C2 in (8.2.9) byabq,aq,q-X, cqx-N, bcq, c- 1q-Y, qy-N, respectively, to obtain the following product formula for the q-Racah polynomials introduced in §7.2: VVn(x;a,b,c,lV;q) VVn(y;a,b,c,lV;q) n (q-n abqn+1 qx-N qy-N c- 1q-X c- 1q-Y. q) (b q, qac -1.) ,q n n '"""' ' , , , , 'm m = (aq, bcq; q) n c m=O ~ (b q q, q, qac -1 ,c -1 ,q -N , q-N ; q) m ,/,. [cq- m , q(cq-m)~, _q(cq-m)~, ca- 1q-m, b- 1q-m, q-m, q-X, X 10'1-'9

1

1

1

(cq-m)2, _(cq-m)2, aq, bcq, cq, cqx+ -m,

q -y , cq x-N , cq y-N b 2N +3 ] y+1-m N -x+1-m N -y+1-m ; q, a q , cq ,q ,q

(8.3.1)

where VVn(x;a,b,c,lV;q) _ -

41>3

[q-n, abqn+1, q-X, cqx-N . ] -N b ,q,q aq,q ,cq

(8.3.2)

is the q-Racah polynomial defined in (7.2.17). This is a Watson-type formula. Two additional Watson-type formulas are given in Ex. 8.l. Letting c ----t 0 in (8.3.1) gives a product formula for the q-Hahn polynomials defined in (7.2.21): Qn(x; a, b, lV; q) Qn(Y; a, b, lV; q) b ) n (-n b n+1 x-N y-N ) = (-aqtqG) ( q;q n q ,a q ,q ,q ;q m (aqx+y)-m (aq;q)n m=O (q,bq,q-N,q-N;q)m

L

X 41>3

[

q-X, q-Y, b- 1q-m, q-m b 2N+3] N-x+1-m N-y+1-m; q, a q . aq,q ,q

(8.3.3)

222

Further applications

To obtain a Watson-type product formula for the Askey-Wilson polynomials defined in (7.5.2) we replace a, b, bl , b2, b3 , Cl, C2 in (8.2.9) by abcdq-l, ab, ae i(} , ae- i (}, ac, de i ¢, de- i ¢, respectively, where x = cos B, y = cos ¢. This gives Pn(x; a, b, c, dlq) Pn(Y; a, b, c, dlq) = (ab, ac, ad, ad, bd, cd; q)n(ad)-n n (q-n abcdqn-l de i() de- i(} de i ¢ de-i¢. q)

x

L

'(q, ad, ad, , bd,, cd, da-

'1

m=O A.

' m qm

')

;q

m

[aq- m /d, q(aq-m /d)~, -q(aq-m /d)~, ql-m /bd, ql-m lcd, q-m,

X 10'1"9

1

1

(aq-m /d)'l, -(aq-m /d)'l, ab, ac, aq/d,

ae i(} , ae- i (} , ae i ¢, ae- i¢ bcq ] ql-me-i() /d, ql-mei(} /d, ql-me-i¢ /d, ql-mei¢ /d; q, ad·

(8.3.4)

When b = aq~ and d = cq~, the 1O¢9 series in (8.3.4) becomes balanced and hence can be transformed to another balanced 1O¢9 via (2.9.1). This leads to a Bateman-type product formula Pn(x; a, aq~, c, cq~ Iq) Pn(Y; a, aq~, c, cq~ Iq)

1

X 1O¢9

[v'

qv'l, 1

v 'l ,

ae- i(} , acq~ ei¢+i(} ,

1

-qv'l, 1

-v'l,

ae i ¢,

aq~ ei ¢,

ce i¢,

ac,

a2q~ ,

1

acq'l,

a 2c2qm, .¢ q'l- m e' /c, 1

2

~:mm+!; q, q]

a ce q

2

ae i (} , acq~ ei¢-i(} , ,

(8.3.5)

where v = a2cei¢q-~. In fact, if we replace a and c by q(2o:+l)/4 and _q(2,6+l)/4, respectively, then this gives a Bateman-type product formula for the continuous q-Jacobi polynomials (7.5.24) which, on letting q ----t 1, gives Bateman's [1932] product formula for the Jacobi polynomials:

t

PAOI ,,6)(x)PAOI ,,6)(y) = (_1)nC8 + 1 )n (-nh(n+a+,8+1)k (x+2 y)k OI OI PA ,,6) (l)PA ,,6) (1) (a+1)n k=o k!(,8+1h

x p(OI,,6) k

(1 +

x Y ) /p(a,,6)(l), x+y k

(8.3.6)

which is equivalent to (8.2.13). For terminating series there is really no difference between the Watson formula (8.2.11) and the Bailey formula (8.2.7) since one can be transformed into the other in a trivial way. However, for the continuous q-ultraspherical polynomials given in (7.4.14), there is an interesting Bailey-type product formula that can be obtained from (8.2.6) by replacing a, b, bl , b2, b3 , Cl, C2 by

8.4 A product formula

223

a 4, a2q~ , ae i() , ae- i () " -a2q~ aeiq, and ae-iq, , respectively'.

_

n

'"'

(q-n a 4qn aeiq, ae-iq, -aei () -ae-i(). q) '

-~ -0 mX 1O¢9

(

[

,

,

,

1

,

'm m

1

2 -a 2. q q , a 2q2 , -a 2q2 " -1 -a "

_q-m,

q

)

m

q( _q-m)~,

_q(_q-m)~

-q -m)!2,

_(_q-m)~

(

,

q~-m la2,

_q~-mla2,

q-m,

,

_a2q~ ,

a2q~,

_q,

ae i () , _q1-m e -i() la,

(8.3.7)

For further information about product formulas see Rahman [1982] and Gasper and Rahman [1984].

8.4 A product formula in integral form for the continuous q- ultraspherical polynomials As an application of the Bateman-type product formula (8.3.5) for the AskeyWilson polynomials we shall now derive a product formula for the continuous q-ultraspherical polynomials in the integral form Cn(X; ,8lq)Cn (y; ,8lq)

(r; ~)n q, q

n

,8-n/2j1 K(x, y, Z; ,8lq)Cn (z; ,8lq) dz,

(8.4.1)

-1

where K(

x

. r-II ) = (q,,8,,8;q)=1 (,8e2i(},,8e2iq,;q)=12 x, y, Z, {J q 2n(,82; q)= W

(z;,8~eiO+iq",8~e-i(}-iq",8~ei(}-iq",8~eiq,-i(})

(8.4.2)

with w(z; a, b, c, d) defined as in (6.3.1) and x = cos e, y = cos ¢. First, we set c = -a in (8.3.5) and rewrite it in the form !

!

!

!

rn(x; a, aq2, -a, -aq2Iq)rn(Y; a, aq2, -a, -aq2Iq)

2q = 11++a a 2n(-q _!)n 2

~ (q-n, a 4qn, -a2q~ ei(}+iq" -a2q~ eiq,-i() , -aq~ e-iq,; q) x

~

m=O X

(q, a2q~, -a2q~, -a 2q, -a3q~ eiq,; q) m

mqm

lOW9 (-a3q-~eiq,;aeiq" -aeiq"aq~eiq"aei(},ae-i(},a4qm,q-m;q,q),

(8.4.3) where

224

Further applications

rn(X; a, b, c, dlq) _ -

4¢3

[q-n, abcdqn-1, aeil:l, ae-il:l . ] b d ,q, q . a ,ac,a

(8.4.4)

The key step now is to use the d = -(aq)! case of Bailey's transformation formula (2.8.3) to transform the balanced 1O¢9 series in (8.4.3) to a balanced 4¢3 series:

(a 2e- 2i ¢, -a3q! ei¢; q) m (a 4 , -aq! e- i¢; q) m

X4¢3

[

a2e2i¢, _q! eil:l+i¢, _q! ei¢-il:l, q-m ] 21""'"1:1 2 1 Z"1:1 "'" 1 2"'" 2;q,q· -a q'2 eZ3

=

(q, a 2 , a 2; q)oo I (a 2e 2i ¢, a 2e- 2iO ; q)oo 12 2n(a 4 ; q)oo X

11

W

(z', aei¢-iO , ae iO - i ¢ , ae iO +i¢ , ae- iO - i¢)

-1

(q, a 2 , a 2; q)oo I (a 2e 2i ¢, a 2e- 2iO ; q)oo 12

x

11

2n(a 4 ; q)oo W

(z', aei¢-iO , ae iO - i ¢ , ae iO +i¢ , ae- iO - i¢)

-1

( -aq~ ei'ljJ, -aq~ e-i'ljJ; q) X

m

( -a2q~ ei¢-iO, -q~ e iO - i ¢; q) m

dz.

(8.4.9)

Substituting (8.4.9) into (8.4.7) and using (2.10.4) we finally obtain 1

1

1

1

rn(X; a, aq'2, -a, -aq'2lq) rn(Y; a, aq'2, -a, -aq'2lq)

=

11

K(x, y, z; a 2 lq) rn(Z; a,

aq~, -a, -aq~ Iq) dz.

(8.4.10)

-1

This yields (8.4.1) if we replace a by f3~ and use (7.4.14). By setting f3 = qA in (8.4.1) and taking the limit q --+ 1, Rahman and Verma [1986b] showed that (8.4.1) tends to Gegenbauer's [1874] product formula

O~(x)O~(y) 0;(1)0;(1) =

11

-1

O~(z) K(x, y, z) 0;(1) dz,

(8.4.11)

where (8.4.12) according as 1 - x 2 - y2 - z2 + 2xyz is positive or negative. Rahman and Verma [1986b] were also able to derive an addition formula for the continuous q-ultraspherical polynomials corresponding to the product formula (8.4.1). This is left as an exercise (Ex. 8.11).

226

Further applications

8.5 Rogers' linearization formula for the continuous q- ultraspherical polynomials

Rogers [1895] used an induction argument to prove the linearization formula

Cm(X; ,6lq)Cn (x; ,6lq)

mi~,n) (q; q)m+n-2d,6; q)m-k(,6; q)n-k(,6; q)k(,62; q)m+n-k k=O

(,62; q)m+n-2k(q; q)m-k(q; q)n-k(q; q)d,6q; q)m+n-k

(1 - ,6qm+n-2k) (1 _ ,6)

X

(8.5.1)

Cm+n- 2k (X; ,6lq)·

Different proofs of (8.5.1) have been given by Bressoud [1981d], Rahman [1981] and Gasper [1985]. We shall give Gasper's proof since it appears to be the simplest. We use (7.4.2) for Cn(x; ,6lq) and, via Heine's transformation formula (1.4.3) ,

',61 ) = (,6e- 2iO ; q) 00 (,6; q)m imO X, q (q,6-1e-2iO; q)oo (q; q)m e

c ( m

x

where x

= cos e.

2cPl (q,6-1,,6-2ql-m;,6-1ql-m;q,,6e- 2iO ) ,

Then, temporarily assuming that Iql

(8.5.2)

< 1,61 < 1, we have

(8.5.3) where A

(,6e- 2iO ;q) 00 (,6; q)m(,6; q)n i(m+n)O m,n - (q,6-1e-2iO; q)oo (q; q)m(q; q)n e . _

(8.5.4)

The crucial point here is that the 4cP3 series in (8.5.3) is balanced and so, by (2.5.1),

8.6 The Poisson kernel for Cn(x; ,8lq)

227

Substituting this into (8.5.3) and interchanging the order of summation, we obtain

min(m,n)

X

L

k=O

X

x

(,8-2 ql-m-n; q)2k (q,8-1 e -2iO)k (,8-1 ql-m-n; q)2k A,

2'/'1

(

(./ -2iO) ql-'(./-1 ,I-'(./-2 q1+2k-m-n.,I-'(./-1 q1+2k-m-n.,q,l-'e,

(856) ..

which gives (8.5.1) by using (8.5.2) and observing that both sides of (8.5.1) are polynomials in x. Notice that the linearization coefficients in (8.5.1) are nonnegative when -1 < ,8 < 1 and -1 < q < 1. For an extension of the linearization formula to the continuous q-Jacobi polynomials, see Ex. 8.24.

8.6 The Poisson kernel for Cn(x; ,8lq) For a system of orthogonal polynomials {Pn(x)} which satisfies an orthogonality relation of the form (7.1.6), the bilinear generating function 00

(8.6.1) n=O is called a Poisson kernel for these polynomials provided that h n = CVn for some constant C > O. The Poisson kernel for the continuous q-ultraspherical polynomials is defined by

Kt(x, y; ,8lq)

=

~ ~~;;;)~(~ ~~i Cn(x; ,8lq)Cn (y; ,8lq)tn ,

(8.6.2)

where It I < 1. Gasper and Rahman [1983a] used (8.5.1) to show that

-qt,8! , qte iHi , qte- iO - i , -t,8! , ,8te- iO - i , ,8teiHi , te iO - i , q,8tei-iO,

tei-iO, q,8teiO - i ,

where x = cos(),y = cos¢ and max(lql, Itl, closely related kernel

(8.6.3)

1(81) < 1. They also computed a

228

Further applications ((3, t 2 ; q) 00 ((32, (3t 2; q)oo

teiO+iq, ,

te-iO-iq, ,

(3te- iO - i q, , (3te iHi q, , (8.6.4) Alternative derivations of (8.6.3) and (8.6.4) were given by Rahman and Verma [1986a]. In view of the product formula (8.4.1), however, one can now give a simpler proof. Let us assume, for the moment, that It(3-~ 1< 1 and 1(31 < l. Then, by (7.4.1) and (8.4.1), we find that, with z = cos'lj.;,

1

(t(3~eiW,t(3~e-iw;q)

1

Lt(x, y; (3lq)

=

-1

K(x, y, z; (3lq) (

1.

1.

00)

t(3-2 e2W , t(3-2 e- 2W ; q

dz

00

(q,(3,(3;q)ool ((3e2iO,(3e2iq,;q)00 12 27r((32; q)oo

By (6.3.9),

x 8 W7 ((3t 2q-1., teiO+iq, , te-iO-iq, , teiO-iq, , teiq,-iO , (3." q (3)

.

(8.6.6)

Formula (8.6.4) follows immediately from (8.6.5) and (8.6.6). It is slightly more complicated to compute (8.6.1). Consider the generating function 00

Gt(z)

=

L

1 _ (3qn 1 _ (3 Cn(z; (3lq)tn

n=O ((3te iW , (3te- iw ; q) 00 (l-(3)(te iw ,te- iW ;q)oo

_

~ ((3qte iW , (3qte- iw ; q) 00

1-(3 (qteiW,qte-iW;q)oo

= 1 _ (3t2 ((3qte 2W , (3qte- iw ; q) 00 • (

)

(te 2w , te- 2W ; q) 00

(8.6.7)

8.7 Poisson kernels for the q-Racah polynomials

229

Then Kt(x, Y; ,8lq) =

11

-1

K(x, y, Z; ,8lq) G 13_1 (z) dz t

2

(1- t 2)(q,,8,,8;q)= I (,8e2il1,,8e2i.~, _>.~, ab q N+r+s+2, qr+s-K' q, q ,

= abq2r+2s+1.

We shall now show that when K

(8.7.7)

= N,

( N-z ) q ; q z-r-s A, (qr+s-z ql+r+s-z. ql+N-z. q abqN+z+2) , , , , . ( abq N+r+s+2.,q) z-r-s 2'1"1 (8.7.8) To prove (8.7.8) it suffices to show that Br,s

=

4¢3

[

a,

1

1

-qa 2 , 1 -a2,

qa2, 1 a2,

b w] ; q,w aq

(wblaq, wlb; q)oo 2'1"1, A, (b b' q,w bl' a,q,w Ib) ( w I aq,w;q ) 00

(8.7.9)

whether or not b is a negative integer power of q, provided the series on both sides converge. Since 1 - aq2k = 1 - qk + qk(l - aqk), the left side of (8.7.9) equals 2¢I(aq,b;w;q,wla)+ (wbla,wlb;q)oo (I ) w a,w;q 00

w(l - b) ( ) 2¢I(aq,bq;wq;q,wlaq) aq 1-w

2¢1

(

I I ) b, bq; wb a; q, w b

8.7 Poisson kernels for the q-Racah polynomials wlb; q)oo + w(l(b)(wbla, I ) aq w aq,w;q

rI-.

(wblaq, wlb; q)oo (w I aq,w;q )00

rI-.

(b b· bl· Ib) q, W a, q, w

2'1-'1,

00

=

(b b· q, W bl· a, q, w Ib)

2'1-'1,

by (1.4.5). Also, for 0: = a the which, by (2.10.4), equals

5¢>4

231

series in (8.7.5) reduces to a

(8.7.10) 4¢>3

series

( M+1-y-s b-1 -1 -y-s. ) _' "( q ,q s (b"(qr+s+y-M) s q (qr M, b"(qr+1; q)s X 4¢>3

[

q-S, qr- y , b-1q-S, C"(-l q M+1- y -s ] l+r-s M+1-y-s b- 1 -1 -y-s; q, q . cq ,q ,,,( q

(8.7.11)

From (8.7.6), (8.7.8) and (8.7.11) it follows that JDz(x,y; a,b,c, a,,,(,lV, AI, lV; q)

20

(8.7.12)

for x = 0,1, ... , lV, y = 0,1, ... , AI, z = 0,1, ... , lV when 0 < q < 1, 0 < aq < 1, 0 ::; bq < 1, 0 < c < aqN and cq ::; "( < qM -1 ::; qN -1. Hence the discrete Poisson kernel (8.7.2) is nonnegative for x, y, z = 0,1, ... ,lV when 0 < q < 1, o < aq < 1, 0 ::; bq < 1 and 0 < c < aqN. If in (8.7.3) we write the sum with lV as the upper limit of summation, replace (q-Z; q)n by (tq-K; q)n and let K ----t 00, it follows from (8.7.6) that Lt(x, y; a, b, c, 0:, ,,(, AI, lV; q) N

=

L tnhn(a, b, c, lV; q)Wn(x; a, b, c, lV; q)Wn(y;

0:,

abo:- 1 , ,,(, AI; q)

n=O

(b q,aq I c;q ) N

x N-x

~ ~

(b a q 2.) ,q 2r+2s (-x q ,cqx-N ,q -y ,,,(q y-M.) ,q r

(abq2,c 1;q)N ~ ~ (abqN+2;q)r+s (q,o:q,q-M, ab,,(qo:-1, cql-s; q)r

-

X

( q x-N ,c -1 q -x ; q) s (1 - cq r-s) A C (_t)r+s (r+s)(2N -r-s+1)/2-rs (q,b q,ac- 1 ) s (1-cq-S ) rs q , q;q " rs

(8.7.13) for x

=

0,1, ... , lV, y

C

=

-

r,s -

4¢>3

0, 1, ... , AI with Ar,s defined in (8.7.5) and \! r+s-N \ \ ! [ A, qA2, -qA2, q \! \! b N+r+s+2

A2,-A2,a q

. t N-r-s ] ,q, q ,

(8.7.14)

where A = abq2r+2s+1. However, by Ex. 2.2, rI-.

4'1-'3

[A,

qA~, A~ ,

-qA~, b- 1 ; q, tb] -A~

,

Abq

(t, Aq; q)oo ( ) bAb.) 2¢>l b ,tb;tq;q,Aq (t, q, q 00 for max(ltbl, IAql)

(8.7.15)

< 1. Use of this in (8.7.14) yields

Gr,s = (t, abq 2r+2s+2; q)N-r-s

2¢>1

(qN-r-s, tqN-r-s; tq; q, abq2r+2s+2) ,

(8.7.16)

232

Further applications

from which it is obvious that er,s 2: 0 for 0 :::; t < 1, r + s :::; N when Combining this with our previous observation that Ar,s equals the expression in (8.7.11) when a = a, it follows from (8.7.13) that

o :::; abq2 < 1.

Lt(x,y;a,b,c,a,,,!,M,N;q) > 0

(8.7.17)

for x = 0,1, ... ,N, y = 0,1, ... ,M, 0 :::; t < 1 when 0 < q < 1, 0 < aq < 1, bq < 1, 0 < c < aqN and cq :::; "! < qM-l :::; qN-l. In particular, the Poisson kernel (8.7.1) is positive for x,y = O,l, ... ,N, 0:::; t < 1 when o < q < 1, 0 < aq < 1, 0 :::; bq < 1 and 0 < c < aqN. For further details on the nonnegative bilinear sums of discrete orthogonal polynomials, see Gasper and Rahman [1984] and Rahman [1982].

o :::;

8.8 q-analogues of Clausen's formula Clausen's [1828] formula

{2Fl [a +ai~ ! ;z]

r

F [ 3

2

2a, 2b, a + b 2a + 2b, a + b +

! ;z

]

,

(8.8.1)

where Izl < 1, provides a rare example ofthe square of a hypergeometric series that is expressible as a hypergeometric series. Ramanujan's [1927, pp. 2339] rapidly convergent series representations of l/n, which have been used to compute n to millions of decimal digits, are based on special cases of (8.8.1); see the Chudnovskys' [1988] survey paper. Clausen's formula was used in Askey and Gasper [1976] to prove that 3

F [-n,n+a+2,!(a+1).1-x] > 0 2 a+1,!(a+3) '2 -

() 8.8.2

when a > -2, -1 :::; x :::; 1, n = 0,1, ... , which was then used to prove the positivity of certain important kernels involving sums of Jacobi polynomials; see Askey [1975] and the extensions in Gasper [1975a, 1977]. The special cases a = 2,4,6, ... of (8.8.2) turned out to be the inequalities de Branges [1985] needed to complete the last step in his celebrated proof of the Bieberbach conjecture. In this section we consider q-analogues of (8.8.1). Jackson [1940, 1941] derived the product formula given in Ex. 3.11 and additional proofs of it have been given by Singh [1959], Nassrallah [1982]' and Jain and Srivastava [1986]. But, unfortunately, the left side ofit is not a square and so Jackson's formula cannot be used to write certain basic hypergeometric series as sums of squares as was done with Clausen's formula in Askey and Gasper [1976] to prove (8.8.2). In order to obtain a q-analogue of Clausen's formula which expressed the square of a basic hypergeometric series as a basic hypergeometric series, the authors derived the formula

a,b,abz,ab/z . { 4¢3 [ b! a q2, -ab! q2,-a b,q,q

]}2

5¢4

a2,b2,ab,abz,ab/z ] [ 2b2 b! b! b; q, q , a ,a q2,-a q2,-a (8.8.3)

8.8 q-analogues of Clausen's formula

233

which holds when the series terminate. See Gasper [1989b], where it was pointed out that there are several ways of proving (8.8.3), such as using the Rogers' linearization formula (8.5.1), the product formula in §8.2, or the Rahman and Verma integral (8.4.10). In this section we derive a nonterminating q-analogue of Clausen's formula which reduces to (8.8.3) when it terminates. The key to the discovery of this formula is the observation that the proof of Rogers' linearization formula given in §8.5 is independent of the fact that the parameter n in the 21>1 series in (7.4.2) is a nonnegative integer. In view of (7.4.2) let

J(z)

= 21>1(a,(3;aql(3;q,zql(3),

(8.8.4)

which reduces to the 21>1 series in (7.4.2) when a = q-n and z = e- 2iIJ . Temporarily assume that Iql < 1(31 < 1 and Izl ::::; 1. From Heine's transformation (1.4.3) , (8.8.5) Hence, if we multiply the two coefficients of zj, we get

21>1

series in (8.8.4) and (8.8.5) and collect the

(8.8.6) where

Ak =

41>3

[

q-k,(3,(3q-kla,a ] (3q-k, (32 q-k la, aql (3; q, q

(8.8.7)

is a terminating balanced series. As in (8.5.5) we now apply (2.5.1) to the series in (8.8.7) to obtain that

2ql(32;q)00 A k -- (aql(3,a (a 2 q1(3 , aq 1(32.) , q 00

8

w:7 (2/(3· (3 2 k+l/(32 -k. 1(3) a ,a,a"a q ,q ,q,q .

41>3

( ) 8.8.8

Using (8.8.8) in (8.8.6) and changing the order of summation we get the formula

f=

J2(z) - ((3z;q)oo (1- a 2q2k l(3)(a 2/(3,a,a,(3;q)k - (zql (3; q)oo k=O (1 - a 21(3)(q, aql (3, aql (3, a 2ql (32; q)k (a 2ql(32;q)zk (Zq)k x (a 2ql (3; q)zk

/3

21>1

[a2q2k+l/(32,ql(3 ] a 2 q2k+l 1(3 ; q, (3z . (8.8.9)

Observe that since the 21>1 series in (8.8.9) is well-poised we may transform it by applying the quadratic transformation formula (3.4.7) to express it as an 81>7 series and then apply (2.10.10) to get the transformation formula

1> [a2q2k+l/(32,ql(3 (3] 2 1 a2q2k+l 1(3 ; q, z [ X 41>3

=

(k+l) (azql(3;q)oo(-al(3z)k q 2 ((3zla; q)oo(aql (3z, azql (3; q)k

] aqk, aqk+! 1(3, -aqk+! 1(3, _ aqk+ 1 1(3 a2q2k+l 1(3, azqk+l 1(3, aqk+l 1(3z ; q, q

234

Further applications

1

X

rI-.

4'f'3

[

1

/3z, zq"2 , -zq"2, -zq ] ; q, q . qz2, o: zq k+l, /3 zq l-k /0:

(8.8.10)

We can now substitute (8.8.10) into (8.8.9) and change the orders of summation to find that

f

J2(Z) = (/3z,o:zq//3;q)oo (0:; q)m(0:2q//3 2; qhm (zq/ /3, /3z / 0:; q)oo m=O (q, o:zq/ /3, o:q/ /3z, 0:2 q/ /3, o:q/ /3; q)m X

qm 6W 5(0:2//3;0:,/3,q-m;q,o:qm+l//32)

+

(0:, 0:2q/ /3, zq, o:zq; q)oo (/3z, zq!, -zq!, -zq; q)m m (o:q/ /3, 0:2q/ /3, zq/ /3,0:/ /3z; q)oo m=O (q, qz2, o:zq, /3zq/o:; q)m q

f

X6W5(0:2 //3; 0:, /3, o:q-m / /3z; q, zqm+l /(3). Summing the above {

6W5

(8.8.11)

series by means of (2.7.1), we obtain the formula

2 (/3z, o:zq/ /3; q)oo 2¢>1 ( 0:, /3; o:q / /3; q, zq / /3 )} = (zq //3 , /3 Z/ 0:,. q) rI-. [0:, o:q/ /3 2, o:q! //3, -o:q! //3, -o:q/ /3 ] X 5'f'4 o:q/ /3, 0:2q/ /3 2, o:q/ /3z, o:zq/ /3 ; q, q (0:, o:q/ /3 2, zq, zq, o:zq/ /3; q)oo 00

+~~~~~=-~~~~~-

(o:q//3,o:q//3,zq//3,zq//3,o://3z;q)oo rI-.

x 5'f'4

[/3z,Zq//3,Zq!,-Zq!,-Zq ] 2 ; q, q , zq, z q, /3zq/o:, o:zq/ /3

(8.8.12)

which gives the square of a well-poised 2¢>1 series as the sum ofthe two balanced 5¢>4 series. By analytic continuation, (8.8.12) holds whenever Iql < 1 and

Izq/ /31 < l. To derive (8.8.3) from (8.8.12), observe that if 0: 0 and (8.8.12) gives

q-n, n

=

=

0,1, ... , then

(0:; q)oo =

J2(Z) (/3z, zql-n//3;q)oo ¢> [q-n,q!-n//3,ql-n//32,_q!-n//3,_ql-n//3 ] = (/3zqn,zq//3;q)oo 5 4 ql-n//3,ql-2n//32, zq l-n//3,ql-n//3z ;q,q _ (/3 2,/32;q)n (.:.)n rI-. [q-n,/32 qn,/3,/3z,/3/z. ] - (/3 , /3.) 5'f'4 /32 , /3 q2,1 /3 q2,1 /3' q, q (8.8.13) ,q n /3 by reversing the order of summation. Since

1

2;q)n (.:.)n rI-. [q-n,/32 qn, (/3z)!, (/3/z)! . J( z ) = (/3 (/3) /3 4'f'3 1 1 , q, q ;q n /3q"2, -/3q"2, -/3 by (7.4.14), it follows from (8.8.13) that rI-. {

4'f'3

l}2

[q-n,/32 qn,(/3z)!,(/3/Z)!. /3 q2,1 /3 q2,1 /3 ,q,q

(8.8.14)

8.8 q-analogues of Clausen's formula

235 (8.8.15)

for n = 0,1, ... , which is formula (8.8.3) written in an equivalent form. Now note that 2¢1(a,~;aq/~;q,zq/~)

(z(aq)!, -z(aq)!, zqa! I~, -zqa! I~; q)oo (zq!, -zq!, zq/~, -azq/~; q)oo X 8 W 7 ( -az/~;

a!, -a!, (aq)! I~, -(aq)! I~, -z; q, -zq),

(8.8.16)

by (3.4.7) and (2.10.1), and set a = a!,b = (aq)!/~ to obtain from (8.8.12) the following q-analogue of Clausen's formula:

_1

}2

(a2z2q, b2z 2q; q2)00 { (2 2b 2 . 2) 8 W7( -abzq 2; a, -a, b, -b, -z; q, -zq) z q,a 2z q,q 00 _ (azq! Ib, bzq! la; q)oo ,j., [ a2, b2, ab, -ab, -abq! . ] 1 1 5¥'4 1 1 1 , q, q (zq'i lab, abzq'i; q)oo a2b2, abq'i, abzq'i, abq'i I z

+

(zq, zq, a 2, b2; q)oo 1 1 1 (abq'i,abq'i,ablzq'i,abzq'i;q)oo 1

1

1 -zq'i, 1 -zq azq'i1 Ib, bzq'i1 la, zq'i,

X 5¢4

[

2

1

3

zq,z q,abzq'i,zq'ilab

;

q, q ,

(8.8.17)

where Iql < 1 and Izql < 1. To see that (8.8.17) is a nonterminating q-analogue of Clausen's formula, it suffices to replace a by qa, b by qb and let q ----+ 1-; then the left side and the first term on the right side of (8.8.17) tend to the left and right sides of (8.8.1) with z replaced by -4z(l- Z)-2 and so, by (8.8.1), the second term on the right side of (8.8.17) must tend to zero. It is shown in Gasper and Rahman [1989] that the nonterminating extension (3.4.1) of the Sears-Carlitz quadratic transformation can be used in place of (8.8.10) to derive the product formula 2¢1

(a,b;c;q,z ) ,j.,

x

6¥'5

2¢1

(a,aq I c;aq I b;q,z )

(az,abzlc;q)oo

= (z, bz I c; q )

00

[a, clb, (aclb)!, -(aclb)!, (acqlb)!, -(acqlb)! ] I I I ; q, q aq b, c, ac b, az, cq bz

+ (a,clb, az, bz,azqlc;q)oo (c,aqlb,z,z,clbz;q)oo ,j.,

x

6¥'5

[z, abzlc, z(ablc)!, -z(ablc)!, z(abqlc)!, -z(abqlc)! ] I I 21 ; q, q , az, bz, azq c, bzq c, abz c (8.8.18)

where Izl < 1 and Iql < 1. This formula reduces to (8.8.12) when a ~, c = aql ~ and z is replaced by zq/~.

= a, b =

236

Further applications

By applying various transformation formulas to the 2¢1 series in (8.8.12) and (8.8.18), these formulas can be written in many equivalent forms. For instance, by replacing b in (8.8.18) by c/b and applying (1.5.4) we obtain

2¢2(a, b; c, az; q, cz/b) 2¢2(a, b; abq/c, az; q, azq/c) ] _ (z,az/b;q)oo " [a,b,(ab)!,-(ab)!,(abq)!,-(ab q)!. - (az, z /b) ' q, q ; q 00 6'1'5 abq / c, c, ab, az, bq / z

+ (a, b, cz/b, azq/c; q)oo (c, abq/c, az, b/ z; q)oo " [z, abz, z(a/b)!, -z(a/b)!, z(aq/b)!, -z(aq/b)! ] ( ) ; q, q , 8.8.19 az, cz/b, aqz/c, zq/b, az 2/b

X 6'1'5

where max(lql, lazq/cl, Icz/bl) < 1. If we replace a, b, z in (8.8.19) by qa, bb, z/(z - 1), respectively, and let q ----+ 1-, we obtain the Ramanujan [1957, Vol. 2] and Bailey [1933, 1935a] product formula

2Fl(a, b; c; z) 2Fl(a, b; a + b - c + 1; z) = 4F3(a, b, (a + b)/2, (a + b + 1)/2; c, a + b, a + b - c + 1; 4z(1 - z)), (8.8.20) where Izl < 1 and 14z(l- z)1 < 1. This is an extension of Clausen's formula in the sense that by replacing a, b, c, 4z(1 - z) in (8.8.20) by 2a, 2b, a + b + ~,z, respectively, and using the quadratic transformation (Erdelyi [1953, 2.11 (2)]) 1

1

2Fl(2a, 2b; a + b + "2; z) = 2Fl(a, b; a + b + "2; 4z(1 - z)),

(8.8.21)

we get (8.8.1). See Askey [1989d].

8.9 Nonnegative basic hypergeometric series

Our main aim in this section is to show how the terminating q-Clausen formula (8.8.3) can be used to derive q-analogues of the Askey-Gasper inequalities (8.8.2) and of the nonnegative hypergeometric series in Gasper [1975a, Equations (8.19), (8.20), (8.22)]. As in Gasper [1989b], let us set 'Y Cl

= q2b,al = qb,a2 = qb eiO,a3 = qbe-iO,b 1 = q2b,b2 = _qb,

= q-n ,C2 = qn+a ,d1 = q2!(a+l) = -

d 2, el

= qH!2 = -e3, x = q, W = 1 ,

in the r = 3, s = t = u = k = 2 case of (3.7.9) to obtain the expansion

8.9 Nonnegative basic hypergeometric series

x 5rP4 [

237

q-j,qj+2b, qb,qbeiO, qbe-iO ] 2b HI HI b ;q,q, q ,q 2 , -q 2 , -q

(8.9.1)

where, as throughout this section, n = 0,1, .... By Ex. 2.8 the 4rP3 series in (8.9.1) equals zero when n - j is odd and equals ( q, q a-2b.,q 2) k 2k(n-2k+b+l/2) (q2n-4k+a+l, q2n-4k+2H2; q2)k q

when n - j = 2k and k = 0,1, .... Hence, using (8.8.3) to write the 5rP4 as the square of a 4rP3 series, we have 5rP4

[

q-n, qn+a, qb, qbeiO, qbe-iO ] 2b I (a+l) I (a+l) b ; q, q ,-q2 ,-q q ,q2

[n/2]

=

~

bIb

1

(_l)n(q-n,qn+a,q +"2,-q +"2;q)n-2k (q, q! (a+I), _q! (MI), qn-2k+2b; q)n-2k

(8.9.2)

(8.9.3) when a ~ 2b > -1 and 0 follows from (8.9.3) that 3

F

2

c > 0, which gives the nonnegativity of the 3F2 series in Gasper [1975a, (8.22)] when a = a + ~,b = ~a + and c = a + 1. Also see Gasper [1989d].

£

8.10 Applications in the theory of partitions of positive integers In the applications given so far we have dealt almost exclusively with orthogonal polynomials which are representable as basic hypergeometric series. These are important results and most of them have appeared in print during the last thirty years. They constitute the main bulk of applications as far as this book is concerned. However, our task would remain incomplete if we failed to mention some of the earliest examples where basic hypergeometric series played crucial roles. The simplest among these examples is Euler's [1748] enumeration p( n) of the partitions of a positive integer n, where a partition of a positive integer n is a finite monotone decreasing sequence of positive integers (called the parts ofthe partition) whose sum is n. To illustrate how p(n) arises in a q-series let us consider a formal series expansion of the infinite product (q; q)~} in powers of q: 00

k=O

00

(8.10.1) where

240

Further applications

n

=

kl . 1 + k2 . 2 + ... + k n . n,

(8.10.2)

p(O) = 1 and, for a positive integer n, p(n) is the number of partitions of n into parts::; n. In the partition (8.10.2) of n there are k m m's and hence 0::; k m ::; n/m, 1 ::; m ::; n. For small values of n, p(n) can be calculated quite easily, but the number increases very rapidly. For example, p(3) = 3, p( 4) = 5, p(5) = 7, but p(243) = 133978259344888. Hardy and Ramanujan [1918] found the following asymptotic formula for large n:

p(n)

~ 4n~ex+ C;) ']

(8.10.3)

Also of interest are the enumerations of partitions of a positive integer n into parts restricted in certain ways such as: (i) PN(n), the number of partitions of n into parts::; N, which is given by the generating function 00

(8.10.4) (q; q)"i/ = 2::PN(n)qn, n=O where n = kl . 1 + k2 ·2+ ... + kN . N has k m m's ; (ii) Pe(n), the number of partitions of an even integer n into even parts, generated by 00

II (1 _ q2k+2)-1

(q2; q2)~;;} =

k=O 00

(8.10.5)

= 2::Pe(n)qn; n=O

(iii) Pdist(n), the number of partitions of n into distinct positive integers, generated by 1

(-q; q)oo =

1

2:: 2:: ... qk ·Hk ·2+ .. 1

k1=O

2

k2 =O

00

=

2::

(8.10.6) Pdist (n )qn, n=O where n = kl . 1 + k2 ·2+ ... + k n . n, 0 ::; k i ::; 1, 1 ::; i ::; n, and (iv) po(n), the number of partitions of n into odd parts, generated by 00

(q;q2)~1= II(1-q2k+l)-1

k=O 00

(8.10.7) Euler's partition identity Pdist(n) = po(n)

(8.10.8)

8.10 Applications in the theory of partitions of positive integers

241

follows from (8.10.6), (8.10.7) and the fact that (-q; q)OO =

(q'l,1 -q'l;1)-1 q = (q; q2);;}. 00

(8.10.9)

Other combinatorial identities of this type can be discovered from q-series identities similar to, but perhaps somewhat more complicated than, (8.10.9). For example, let us consider Euler's [1748] identity involving the pentagonal numbers n(3n ± 1)/2: 00

n=-oo 00

00

n=l n=l which is given in Ex. 2.18. A formal power series expansion gives 00

k=O

=

1

1

k1=O

k2 =O

L L ... (_I)kl+k2+··qkl·Hk2·2+ .. ,

(8.10.11)

which differs from the multiple series in (8.10.6) only in the factor (-I)kl+k 2+··. This factor is ±1 according as the partition has an even or odd number of parts. Denoting these numbers by Peven (n) and Podd (n), respectively, we find that 00

(q; q)oo = 1 +

L

[Peven(n) - pOdd(n)] qn.

(8.10.12)

n=l

From (8.10.10) and (8.10.12) it follows that _{(-I)k Peven (n) -

Podd (n)

forn=k(3k±I)/2,

-

(8.10.13) otherwise. Thus Euler's identity (8.10.10) expresses the important property that a positive integer n which is not a pentagonal number of the form k(3k ± 1)/2 can be partitioned as often into an even number of parts as into an odd number of parts. However, if n = k(3k ± 1) /2, k = 1,2, ... , then Peven (n) exceeds Podd (n) by (-I)k. See Hardy and Wright [1979], Rademacher [1973] and Andrews [1976, 1983] for related results. Two of the most celebrated identities in combinatorial analysis are the socalled Rogers-Ramanujan identities (2.7.3) and (2.7.4) which, for the purposes of the present discussion, we rewrite in the following form:

o

1+

L 00

n2

-qn=l (q;q)n

1

= --------

fi (1- q5k+ 1) (1- q5k+4)

k=O

(8.10.14)

242

Further applications 00

1+

n=l

1

qn(n+l)

L ( )

=

TI (1 -

q; q n

q5k+2)

(1 -

(8.10.15) q5k+3)

k=O

It is clear that the infinite product on the right side of (8.10.14) enumerates the partitions of a positive integer n into parts of the form 5k + 1 and 5k + 4, while that on the right side of (8.10.15) enumerates partitions of n into parts of the form 5k + 2 and 5k + 3, k = 0,1, .... Following Hardy and Wright [1979] we shall now give combinatorial interpretations of the left sides of the above identities. Since

k2

=

1 + 3 + 5 + ... + (2k - 1),

we can exhibit this square in a graph of k rows of dots, each row having two more dots than the lower one. We then take any partition of n-k 2 into at most k parts with the parts in descending order, marked with x's and placed at the ends of the rows of dots to obtain a partition of n into parts with minimal difference 2. For example, when k = 5 and n = 32 = 52 + 7 we add 4 x's to the top row, 1 each to the 2nd, 3rd and 4th rows, counted from above. This gives the partition 32 = 13 + 8 + 6 + 4 + 1 displayed in the graph below.

•

•

•

•

•

•

•

•

•

•

•

•

•

•

x

•

•

•

•

x

•

•

x

•

x

x

x

x

• The identity (8.10.14) states that the number of partitions of n in which the differences between parts are at least 2 is equal to the number of partitions of n into parts congruent to 1 or 4 (mod 5). Observing that k(k

+ 1) = 2 + 4 + 6 + ... + 2k,

a similar interpretation can be given to the left side of (8.10.15). Since the first number in the above sum is 2, one deduces that the partitions of n into parts not less than 2 and with minimal difference 2 are equinumerous with the partitions of n into parts congruent to 2 or 3 (mod 5). For more applications of basic hypergeometric series to partition theory, see Andrews [1976-1988], Fine [1948, 1988], Andrews and Askey [1977], and Andrews, Dyson and Hickerson [1988]. Additional results on RogersRamanujan type identities are given in Slater [1951, 1952a], Jain and Verma [1980-1982] and Andrews [1975a, 1984a,b,c,d].

8.11 Representations of positive integers as sums of squares

One of the most interesting problems in number theory is the representations of positive integers as sums of squares of integers. Fermat proved that all

8.11 Representations of positive integers as sums of squares

243

primes of the form 4n + 1 can be uniquely expressed as the sum of two squares. Lagrange showed in 1770 that all positive integers can be represented by sums of four squares and that this number is minimal. Earlier in the same year Waring posed the general problem of representing a positive integer as a sum of a fixed number of nonnegative k-th powers of integers (positive, negative, or zero) with order taken into account and stated without proof that every integer is the sum of 4 squares, of 9 cubes, of 19 biquadrates, 'and so on'. More than 100 years later Hilbert [1909] proved that all positive integers are representable by s kth powers where s = s(k) depends only on k. For an historical account of the Waring problem, see Dickson [1920]' Grosswald [1985], and Hua [1982]. To illustrate the usefulness of basic hypergeometric series in the study of such representations we shall restrict ourselves to the simplest cases: sums of two and sums of four squares, where it is understood, for example, that n = xi + x§ = yi + y§ are two different representations of n as a sum of two squares if Xl -=1= Yl or X2 -=1= Y2· Let r2k (n) be the number of different representations of n as a sum of 2k squares, k = 1,2, .... We will show by basic hypergeometric series techniques that, for n ~ 1,

r2(n) = 4(d l (n) - d3 (n)),

(8.11.1)

L

(8.11.2)

r4(n) = 8

d,

dln,4id

where di(n), i = 1,3, is the number of (positive) divisors of n congruent to i (mod 4) and the summation in (8.11.2) indicates the sum over all divisors of n not divisible by 4. The numbers 4 and 8 in (8.11.1) and (8.11.2), respectively, reflect the fact that r2(1) = 4 since 1 = 02 + (±1)2 = (±1)2 +0 2, and r4(1) = 8 since 1 = 02 + 02 + 0 2 + (± 1)2 = 0 2 + 02 + (± 1) 2 + 0 2 = 02 + (± 1) 2 + 02 + 02 = (±1)2 + 02 + 0 2 + 02. Both of these results were proved by Jacobi by means of the theory of elliptic functions, but the proofs below are based, as in Andrews [1974a], on the formulas stated in Ex. 5.1, 5.2, and 5.3. Combining Ex. 5.1 and 5.2 we have (8.11.3) However, the bilateral sum on the left side is clearly a generating function of (-1 )nr2 (n) and so it suffices to prove that 00

00

"[dl(n) - d3 (n)](-qt ~

(-1 )nqn(n+l)/2

(8.11.4) l+qn n=l n=l By splitting into odd and even parts and then by formal series manipulations, we find that 00 (-1 )nqn(n+l)/2 00 qr(2r+l) 00 q(m+l)(2m+l)

L

n=l

1+

qn

=

L r=l

1+

00

00

r=l m=r

q2r -

= "

~

L

m=O

1+

q2m+l 00

00

m=Or=m+l

244

Further applications

(Xl

(Xl

m=Or=l (Xl

=L

(Xl

L(-lr (q(4m+l)r _ q(4m+3)r)

m=Or=l (Xl

=

L( -q)n(d (n) 1

d3 (n)),

(8.11.5)

n=l

which completes the proof of (8.11.4). To prove (8.11.2) we first replace q by -q in Ex. 5.1 and 5.3 and find that

nqn

(Xl

= 1 + 8 ~ 1 + (_q)n '

(8.11.6)

where the last line is obtained from the previous one by expanding (1 + (_q)n)-2 and interchanging the order of summation. Now,

-- L (Xl

n=l

nqn -+ 1 - qn

L

even

n~2

nq

nIl ----1 + qn 1 _ qn (

)

(8.11. 7) Thus, (8.11.8)

Exercises

245

Since T4(O) = 1, this immediately leads to (8.11.2). For a direct proof of (8.11.8) based only on Jacobi's triple product identity (1.6.1), see Hirschhorn [1987]. Ex. 5.1 and 5.4 can be employed in a similar manner to show that

T8(n) = 16( _l)n~) _1)dd 3,

n ~ 1.

(8.11.9)

din

See Andrews [1974a]. Some remarks on other applications are given in Notes §8.11.

Exercises

8.1 Prove for the little q-Jacobi polynomials Pn(x; a, b; q) defined in (7.3.1) that

(i)

(ii)

Pn(x;a,b;q)Pn(y;a,b;q) n (b ) n (q-n abqn+1. q) m = (-bq)-n q -(2) q;qn 'm(_b yq 2)m q (2) (aq; q)n m=O (q, bq; q)m

L

(q-m abqm+1. q) (' ) ' k ( xq )k k=O q, aq; q k m

X '""' ~

'

A. ( qk-m ,xq;; b 0 2'/-'1

q, b-1 y -1) .

8.2 Derive the following product formula for the big q-Jacobi polynomials defined in (7.3.10):

246

Further applications

8.3 Prove that (abq,aq;q)n(1-abq2n+l) (t)n L ( b ' ) (1 _ b) Pn(qX; a, b; q)Pn(qY; a, b; q) n=O q, q, q n a q aq 00

= (t, abq2; q)oo L 00

r=O X 2¢1

(-X

min(x,y)

L

-Y

)

-s

q ,q ; q sa q(x+y)(r+s) (q, bq; q)s (q, aq; q)r

s=O

(0,0; qt; q, abq2r+2s+2) q-2rs-s 2 t r +s ,

and that this gives the positivity of the Poisson kernel for the little q-Jacobi polynomials for x, y = 0,1, ... , 0:::; t < 1 when 0 < q < 1,0 < aq < 1 and O1

(qr+s-z, q1+r+s-z; q1+N -z; q, abqN +Z+2)

= 0,1, ...

,N, Y

= 0,1, ...

o < aq < 1, 0 :::; bq < 1 and N

,M, z

= 0,1, ...

2 O.

,N, when 0


- ~ , ). ;::: 0, (Gasper [1989b])

°
-(xqk)-l

=

(bxq; qh (x-I; q)oo (a/x)k, (q; qh (q; q)oo

and so a suitable comparison function for asymptotic purposes is

f

(bxq; qh (x-\ q)oo (a/x)k k=O (q;q)k (q;q)oo l+xtqk

= (x-\ q)oo (q;q)oo

f

==0

(abq=+\ q)oo (-xt)=. (aq=/x;q)oo

(9.2.4)

9.2 The little q-Jacobi polynomials

261

Combining (9.2.4) and the left side of (9.2.1) we find, by Darboux's theorem, as extended by Ismail and Wilson [1982]' that

(x-I. q) (n) Pn(x;a,b;q) "-' ( ') 00 (_x)n q- 2 , aq; q 00

x

#

0,I,q,q2, ... ,

uniformly for x, a and b in compact sets. Now we shall reconsider the Poisson kernel for the little q-Jacobi polynomials given in Ex. 8.3, which has the right positivity properties but suffers from the flaw that it does not directly lead to the known q ----+ 1- limit:

f

n=O

=

(2n + a + (3 + 1) (a + (3 + 1 )nn! t n p(a,(3) (cos 2(})p(a,(3) (cos 2c/J) (a + (3 + 1)(a + l)n((3 + l)n n n

(1

1- t

+ t)a+(3+2 F4

[a +

(3 + 2 2 '

a+

(3 + 3 2 ;a

+ 1, (3 + 1;

a2

b2 ]

/1,2' /1,2

'

(9.2.5)

Hi!

where a = sin(}sinc/J, b = cos () cos c/J, /1, = +c~), 0:::; (},c/J:::; 7[, 0 < t < 1, see Bailey [1935, p. 102]. First, use the product formula Ex. 8.1(i) to obtain the following expression

1 - abq2n+l (abq, aq; q)n 1_ b (b .) (t/aq)n Pn(x; a, b; q) Pn(Y; a, b; q) n=O a q, q, q n

=L 00

Kt(x, y)

L L

min(x,y) (

00

=

b 2) (-X -y) a q ;q 2r+2s q ,q str+sa-Sq(x+y)(r+s)-s2-2rs (q,aq;q)r(q,bq;q)s

r=O

s=O

x

1 - abq2(n+r+s)+1 (abq2r+2s+\ q)s (~) -t n

00

~

1- abq2(r+s)+1

(q; q)n

q

(

).

(9.2.6)

What is needed to compute now is a series of the form

f

n=O

1 - aq2n (a: q)n qG) (_t)n == ht(a), say. I-a (q,q)n

(9.2.7)

Without the qG) factor the computation is trivial via the q-binomial theorem (1.3.2). But with this factor the best one can do seems to be to consider this as the b ----+ (Xl limit of the very-well-poised series 4 W3(a;

b; q, t/b)

which, by (3.4.8), transforms to

(t,aq 2t 2;q)00 SW7 ( aqt; ()1 2 ,-qa 1 2 ,bqt;q,t / b) aq 2,_ ()1 aq 2,qa 1 (qt2, atq 2) ; q 00 =

(1 _ t) (-qta~, -:q~ a~, t(aq)~ /~, tqa~ /b; q)oo (-tq'2, -tq, t/b, atq'2 /b; q)oo 1

X

1

1

1

1

1

1)

SW7 ( atq'2/b;tq'2,(aq)'2,qa'2,-a'2/b,-(aq)'2/b;q,tq'2 , (9.2.8)

262

Linear and bilinear generating functions

from (2.10.1). Since 0

< t < 1, it follows by letting b ----;

ht(a) = (1 _ t)

that

00

(-tqa~ 'l-tq~ d; q)oo (-tq'l, -tq; q)oo

,/-. X 3'1'2

[(aq)~,qa~,tq~.q tq~] 1 3 1" -qta'l,-tq'la'l

(9.2.9)

•

So, if we replace a and b in (9.2.6) by qa and q(3, respectively, and use (1.6) and (1.25), then Kt(x, y) of (9.2.6) can be written in a more suggestive form

Kt(x,y)

=

(1- t) 1

1

(-tq'l; q'l )a+(3+2 00

min(x,y) (q~(a+(3+2), q~(a+(3+3), _q~(a+(3+2), _q~(a+(3+3);

(_ tq~(a+(3+3) "_tq~(a+(3+4). q) r+s

xL L r=O s=O x

q)r+s

(q-X, q-Y; q)s (t qx+y)r+s q-s 2-2rs-2s ~~~~~~~~~~----­

(q, qa+l; q)r(q, q(3+1; q)s qr+s+~ (a+(3+2) qr+s+~ (a+(3+3) tq~ [ _tqr+s+~(a+(3+3), _tqr+s+~(a+(3+4) ; q, tq2 .

1]

X 3¢2

(9.2.10)

It is obvious that the q ----; 1- limit of (9.2.10) is indeed (9.2.5) and that the expression on the right side of (9.2.10) is nonnegative when 0 < t < 1 and a,j3 > -1.

9.3 A generating function for Askey-Wilson polynomials There are many different generating functions for the Askey-Wilson polynomials Tn(X; a, b, e, dlq) defined in (8.4.4), of which G t (x; a, b, e, d I) q

(abed/q; =~ ~ (.) q)n t nTn (X; a, b, e, d I) q, n=O

q, q n

It I < 1,

(9.3.1)

is one of the simplest. By Ex. 7.34,

Gt(x;a,b,e,dlq)

= D-1(B) x

3¢2

r

e- iO / d (dueiO,due- iO , abedu/:; q)oo JqeiO /d (dau/q, dbu/q, deu/q, q)oo

q , be, q/u ] [ abed/ ad, abedu/q ; q, adut/q dqu,

(9.3.2)

where D(B) is defined in Ex. 7.34. However, by (3.4.1),

,/-. [abed/q,be,q/u. adut/] 3'1'2 ad, abedu/q ,q, q

= (abedt/q;q)oo (t; q)oo

¢ [(abCd/q)~, -(abed/q) ~, (abed) ~, -(abed) ~,adu/q .

x

5 4

ad, abedu/q, abedt/q, q/t

(abed/q, adt, abcdut/q, adu/q; q)oo (ad, abedu/q, adut/q, l/t; q)oo

+~~~~~--~~--~~~

] ,q, q

9.3 A generating function for Askey-Wilson polynomials

'"

[

X 5'1'4

263

] t(abcdjq) "21 , -t(abcdjq) "21 , t(abcd) "21 , -t(abcd) "21 ,adutjq . j 2j ,q, q . adt,abcdut q,abcdt q,qt (9.3.3)

Substituting this into (9.3.2), using (2.10.18) and simplifying the coefficients, we find that

Gt (x,. a, b,c, dl) q

=

(abcdtjq; q)oo () t; q 00

¢ [(abcdjq)~, -(abcdjq) ~, (abcd) ~, -(abcd) ~ ,aei (}, ae- i (} . ] 5 ab, ac, ad, abcdtjq, qjt ,q, q

x

6

+

(abcdj q, abt, act, adt, ae i (} , ae- i(); q)oo (ab, ac, ad, ate i (} , ate- iO , t- I ; q)oo

x

6'1'5

'" [t(abcdjq)~, -t(abcdjq)~, t(abcd)~, -t(abcd)~, ate iO , ate- iO . ] abt, act, adt, abcdt 2 jq, qt ,q, q . (9.3.4)

The generating function given in Ex. 7.34 is a special case of (9.3.4). A more difficult problem is to evaluate the sum of the series

Ht(x;a,b,c,dlq) =

f

n=O

((a~q))ntn Tn(x;a,b,c,dlq), q, q

(9.3.5)

n

where a is an arbitrary parameter. For a = ab this would, in particular, give a q-analogue of the generating function of Jacobi polynomials given in the previous section. Note that

Ht(x)

== Ht(x; a, b, c, dlq) ~ (abcdjaq; q)m

(a; q)oo

= (a bcdj q,.) q

~

00

m=O

(.) q, q m

a

m

( I ) ( ) Gtqrn x; a, b, c, d q. 9.3.6

So, use of (9.3.4) in (9.3.6) gives

Ht(x)

=

(a, abcdtjq; q)oo (t, abcdjq; q)oo X

'" 6'1'5

f m=O

(t, abcdjaq; q)m am (q, abcdtjq; q)m

[(abcdjq)~, -(abcdjq)~, (abcd)~, -(abcd)~, ae iO , ae- iO . a b,ac, a d ,abcdt qm-I , q I-mjt

+

(a, abt, act, adt, ae iO , ae- iO ; q)oo (ab, ac, ad, ate iO , ate- iO , t- I ; q)oo

x

~--~~~~~--~~~~

x

6'1'5

f

m=O

] ,q, q

(abcdjaq; q)m (q; q)m

. . (m+I) (ate,O,ate-,O;q)mq 2 (-at)m (abt, act, adt, qt; q)m

'" [tqm(abcdjq)~, -tqm(abcdjq)~, tqm(abcd)~, -tqm(abcd)~, abtqm, actqm, adtqm, tqm+ 1 , (9.3.7)

Linear and bilinear generating functions

264

Since, by (1.4.5) 2¢1(abed/aq, tq-n; abedtqn-\ q, aqn) =

(atqn, abedqn-l; q)oo (-n / n-l n) ( bd 1) 2¢1 q ,abed aq; abedq ; q, atq , (9.3.8) aqn, a e tqn- ; q 00

the first term on the right side of (9.3.7) becomes

f

(a, abed/ aq, ae iO , ae- iO ; q)nqG) (-at)n (t; q)oo n=O (q, ab, ae, ad, at, q/t; q)n

(at; q)oo

(9.3.9) Interchange of the order of summation followed by the use of (1.4.5) and simplification in the second term of (9.3.7) gives

(a, at 2 , abedt/q, abt, act, adt, ae iO , ae- iO ; q)oo (ab, ae, ad, at, abedt 2/q, t-I, ate iO , ate- iO ; q)oo x

f

(abed/aq, r\ q)n (at2t (q,abedt/q;q)n

n=O rI.

x 7'1'6

[at, t( abed/ q)! , -t( abed/ q)! , t( abed)! , -t( abed)! , ate iO , ate- iO . ] 2 1 1 ,q, q . abt, act, adt, at ,abedtqn- ,tq -n (9.3.10)

In the special case b = aq!, d = eq!, e --+ -e and a = ab = a 2q!, both 7¢6 series above become balanced 4¢3 's, which along with their coefficients, can be combined via (2.10.10). Denoting the sum of (9.3.9) and (9.3.10) in this combination by Gt(x; a, elq), we get Gt(x; a, elq) (a 2t a2tq! a 2ete iO a 2ete- iO . q) (aetq! , a 3 etq! ,ateiO , ate- iO ; q)oo ,

X

,

,

,(Xl

ae, aeq !2 ,e2 q - !2, ae iO ,ae -iO ; q ) n (a 3 etq!2 ; q) 2n (2 2 !)n 1 1 . 1. a t q2 n=O (q, a 2t, a 2tq2, a 2etq2 e'O, a 2etq2 e-'O; q)n(a 2e 2; qhn 00 "'"'

(

L...J

x 8 W7 ( a 3 etq2n-!; atq! /e, aeqn, aeqn+!, aqneiO, aqne-iO; q, act). (9.3.11) Unfortunately, neither the 8 W 7 series inside nor the outside series over n can be summed exactly except in the limit q --+ 1-. If we replace a and e by q!(o+!) and qH6+!), respectively, in (9.3.11), then lim G t (x;q!(o+!),q!(l3+!)lq)

q-->l -

9.4 A bilinear sum for the Askey-Wilson polynomials I

= (1 _ t)i3+ 1R-a-i3-1 ~ ~

n=O

265

((3)n [2t2 (1 - x)] n n! R2

x F [n+~(a+(3+1),n+~(a+(3+2).2t(1-X)] ( ) 2 1 a + (3 + 1 + 2n ' R 2 · 9.3.12 By the summation formula Erdelyi [1953 Vol. 1. 2.8(6)] we can evaluate the above 2F1 series, and then do the summation over n in (9.3.12) by the binomial theorem (1.3.1) to obtain the well-known generating function Szego [1975, (4.4.5)] for the Jacobi polynomials.

9.4 A bilinear sum for the Askey-Wilson polynomials I

We shall now compute the sum

F(x, ylq) :=

~ (abed)~: !~~~/g; q)n (abed/ fg)nknTn(X)Tn(Y),

(9.4.1)

where Tn(X) is the Askey-Wilson polynomial given in (8.4.4),

k n = K,(a, b, e, dlq)

(111 [Tn(xWw(x)dx) -1,

(9.4.2)

and f and 9 are arbitrary parameters such that the series in (9.4.1) has a convergent sum. By Ex. 7.34,

Tn( cos B)

= B-1 (B) lqe-i8/b qei8/b

Tn(COS¢)

= C- 1(¢) l

(buei!i, bue-i!i, abedu/q; q)oo(ed, q/u; q)n (bau/q, beu/q, bdu/q; q)oo(ab, abedu/q; q)n x (abu/q)ndqu, (9.4.3)

qe - i1> /e ( i¢ . ) 00 (bd ,q /v,q . )n eve ,eve -i¢ ,abedv / q,q qei1>/e (eav/q, ebv/q, edv/q; q)oo(ae, abedv/q; q)n x (aev/q)ndqv, (9.4.4)

where

B(B) = -

iq(l - q) 2b (q,ae,ad,ed;q)ooh(cosB;b)w(cosB;a,b,e,dlq),

(9.4.5)

iq(l - q) (q,ab,ad,bd;q)ooh(cos¢;e)w(cos¢;a,b,c,dlq). 2e

(9.4.6)

and

C(¢)

=-

Hence

F(x, ylq)

=

B-1(B)C-1 (¢) lqe-i8/b (buei!i, bue-i!i, abcdu/q; q)oo qeie /b (bau/q, beu/q, bdu/q; q)oo i qe - 1> /e (eve i¢, eve- i¢, abedv/q; q)oo l X qei1>/e (eav/q,ebv/q,cdv/q;q)oo x 8 W7(abcd/q; ad, f, g, q/u, q/v; q, ab2e2duv/ fgq2)d q ud q v. (9.4.7)

266

Linear and bilinear generating functions

However, by (2.10.10) the above

SW7

equals

(abed, be/ j, be/ g, abed/ j g; q)oo (be, abed/ j, abed/ g, be/ jg; q)oo

x

4¢3

ad,j,g,abeduv/q2 ] [ abedu/q, abedv/q, qjg/be; q, q

(abed, ad, j, g, abeduv/q2, ab 2e2du/ jgq, ab 2e2dv/ jgq; q)oo (be, abed/j,abed/g, abedu/q, abedv/q, ab 2e2duv/jgq2, jg/be; q)oo

+~~--~~~~--~~----~~~--~~~~=-~-

be/j,be/g,abed/jg,ab2e2duv/jgq2

]

x 4¢3 [ ab2e2du/ jgq, ab2e2dv/ jgq, beq/ jg ; q, q ,

(9.4.8)

which, when substituted into (9.4.7), leads to the sum of two terms, say, F Fl + F 2 , where F1(x, ylq)

=

=

(abed, be/ j, be/g, abed/ jg; q)oo B-l(B)C-1(¢) (be,abed/j,abed/g,be/jg;q)oo ~ (ad, j, g; q)n n q n=O (q, qjg/be; q)n

x~

l l

qe - W /b (bue i() , bue-i(} , abeduqn-l; q)oo

qeiO /b (bau/q, beu/q, bdu/q; q)oo i qe - ¢/e (eve i , eve- i , abedvqn-l, abeduvq-2; q)oo x qei¢ /e ( eav/ q, ebv / q, edv / q, abeduvq n-2) ; q 00 dqv dqu, X

(9.4.9) and F 2(x,ylq)

=

(abed,ad,j,g;q)oo B-l(B)C- 1(¢) (be, abed/ j, abed/ g, jg/be; q)oo ~ (be/ j, be/g, abed/ jg; q)n n q n=O (q, beq/ jg; q)n

x~

X

X

l l

qe - iO /b (bue i() , bue- i() , ab 2e2duqn-l / j g; q)oo

qeiO /b (bau/ q, beu/ q, bdu/ q; q)oo qe - i¢/e (eve i , eve- i , ab 2e2dvqn-l / jg, abeduv/q2; q)oo qei¢/e (eav/q, ebv/q, edv/q, ab2e2duvqn-2 / jg; q)oo

x dqv dqu

(9.4.10)

with x = cos B, y = cos ¢. The q-integral over v on the right side of (9.4.9) can be expressed as a terminating S¢7 series via (2.10.19), which can then be transformed to a balanced 4¢3 series that can be transformed back into a different S¢7 series. The final expression for this q-integral turns out to be C(¢) (dei, de- i , bdu/q; q)n (ad, bd, d/a; q)n X S W 7 (aq-n /d; ae i , ae- i , ql-n /bd, q-n, abu/q; q, q2 / adu). (9.4.11)

9.4 A bilinear sum for the Askey-Wilson polynomials I

267

The q-integral over u in (9.4.9) then has the form

l

qe -

iB

qe iB /b

/b

(bue iO , bue- iO , abeduqn-\ q)oo d (bauqk-l, beu/q, bduqn-k-l; q)oo qU,

which, by (2.10.18), gives

B(()) (ae iO , ae- iO ; q)k(de iO , de- iO ; q)n-k. (ad; q)n(ae; qh(ed; q)n-k

(9.4.12)

Substitution of (9.4.11) and (9.4.12) into (9.4.9) then gives F ( 1 X,

I) Yq

(abed, be/ f, be/ g, abed/ f g; q)oo

= (be, abed/ f, abed/g, be/ fg; q)oo x

L (f "9 de iO ,de- iO ,dei ,de-i.,q)n qn 00

n=O

x

10

(q, ad, bd, ed, d/a, qfg/be; q)n

W.9 (aq-n/d· ae iO , ae- iO , ae i , ae- i , ql-n/bd , q1-n/ ed " q-n. q'ad beq ) . \ '

(9.4.13) The q-integral over v in (9.4.10) equals

C ¢ (abdue- i /q, ab 2 edq n e- i / fg; q)oo ( ) (ab 2 eduqn-1 e -2 / fg, abde- 2; q)oo x 8 W 7 (abde- i /q; ae- i , be- i , de- i , q/u, fgq-n /be; q, ab 2 eduqn- 1ei / fg), (9.4.14) which is a bit more troublesome than the previous case because the 8 W 7 series is nonterminating unless fg/be is of the form q-k, k = 0,1, ... , which cannot be the case because of the factor (be/fg;q)oo in the denominator of F 1(x,ylq) and of the factor (fg/be; q)oo in the denominator of F 2 (x, ylq). So either we split up this 8 W 7 series into a pair of balanced 4¢3 series via (2.10.10) and get bogged down in a long and tedious computation, or seek an alternative shorter method. In fact, by (6.3.9) the expression in (9.4.14) can be written as 2 C(rI.) (q ae i , be i , de i ' ab cqn '1-', fg'

b2 cdqn abcdqn fg , fg'

abu/q , bdu/q , adu/q·, q) 00

assuming, for the time being, that (9.4.16)

268

Linear and bilinear generating functions

If we make the further assumption that ad = be, then the q-integral over u reduces to

j

qe- iO /b

qe iO /b

(bue iO , bue- iO , ab 2c2duqn-1 II g; q)oodqu 2 1 (ab eduqn- ei Ilg, q-1 (abd) !uei'I/J-!i, q-1(abd)!ue- i'I/J-!i; q)oo'

which sums to

(9.4.17)

x h(z; eiO-!i(adlb)!, e-iO-!i(adlb)!) . So the q-integral in u over the expression in (9.4.14) equals

B( 8)C( ¢)( q, ae i , bei , ee- i , de i ; q)oo 1(ae iO , ee iO , de iO ; q)oo 12 2n(ab,ae, ad, ad, bd, cd; q)oo (abedqn Ilg, ab 2eqn Ig, b2edqn Ilg; q)oo x (abedqnei+iO Ilg, abedqnei-iO Ilg; q)oo x

h(z;l,-l,q!,-q!)

1

/

-1

h (z; (abl d)! d i , (adlb)! e!i, (bdl a)! e!i ,ae-'"1> ;q, bcq ad)' (ii) From (i) deduce the projection formula m

p:::,b)(x;q) = Lg(n,m)p~o:,f3)(x;q), n=O where p:::,b) (x; q) is the continuous q-Jacobi polynomial defined in (7.5.25), and

(Rahman [1985])

278

Linear and bilinear generating functions

9.18 In (9.5.2) and (9.5.3) take ad = be and Wn =

(ex,(3,"(,o;q)n (be, be, beql f, ex(3"(of Ib 3e3; q)n'

--,----------=-------=--'--------"-'----,---0----=----,-----

Also, in (9.6.2) and (9.6.3) take ad = be and

an

=

(exJlbe,(3J1be,,,(Jlbe,ofl be;q)n n q (ex(3"(oj2 Ib 4 e4 ; q)n .

Choosing appropriate multiples of (9.5.1) and (9.6.1), show that

f

(1 - b2e2q2n-l) (b 2e2q-l, ab, ae, ex, (3, ,,(, 0; q)n (ex(3"( f Ibe; qhn -0 (1 - b2e2q-l) (q be2a- 1 b2ea- 1 f3'Yf a'Yf af3f af3'Y8f. q) (b 2e2. q)2 n'" be ' be ' be ' b3 e 3 , n ,n b2e2qn Of) (f)n ( ex(3"( fbeq2n-l ; ex(3"(f X SW7 b3e3 '-o-,ex qn ,(3qn,,,(qn;q, be a2 x rn(x;a,b,e,bea-1Iq) rn(y;a,b,e,bea-1Iq) (exf Ibe, (3 f Ibe, "( f Ibe, ex(3"( f Ibe; q)oo (ex(3f Ibe, ex"(f Ibe, (3"(f Ibe, f Ibe; q)oo 00

x

~

(ex, (3, ,,(, 0; q)nl(bea-1eil:l, bea-1e i¢; q)nl 2qn (q, be, be, b2ea-1, be2a-1, bea- 2, ex(3"(of Ib 3e3, beql f; q)n

X 10 W g ( a 2q-n Ibe;

aql-n Ib 2e, aql-n Ibe 2, q-n, aeil:l, ae-il:l, ae i¢, ae- i¢; q, q)

(ex, (3, ,,(, 0, ex(3"( f Ibe, ex(3"(oP Ib 4 e4 , f, af Ie; q)oo + (ex(3 f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ab, ae, be, b2el a; q)oo x (aeil:l, bei¢, befeil:l la, efe i¢; q)0012 . 2 (ale, be f I a, bel f, ex(3"(o f Ib 3e3; q)oo 1(Je'CO+¢), fe,CI:I-¢); q)oo 12 X (exJlbe, (3f Ibe, ,,(JIbe, of Ibe, be2f la; q)n 1(JeiCO+¢) ,jeiCO+¢); q)nl 2qn n=O (q, f, qf Ibe, af Ie, ex(3"(oj2 Ib 4 c4; q)n (efe i¢, befeil:l fa; q)nl 2 1

f

1

x

10 W g(be 2fqn-l

fa; fqn, befqn-l, efqn la, eeil:l, ee-il:l, beei¢la, bee- i¢ la; q, q)

(ex, (3, ,,(, 0, ex(3"( f Ibe, f, ef I a, ex(3"(oP Ib 4 e4 ; q)oo (ex(3f Ibe, ex"( f Ibe, (3"( f Ibe, of Ibe, ae, be, b2ea- 1, be2a- 1; q)oo 1(eeil:l, beei¢ I a, bfeil:l, afe i ¢; q)oo 12 x (cia, abf, bel f, ex(3"(o f Ib 3e3; q)oo 1(JeiCO+¢), feiCI:I-¢); q)oo 12

+--,-------,-----~-~--,-------,-----~~--,-----~---,-----~~~~~~--

~ (exf Ibe, (3f Ibe, "( f Ibe, of Ibe, abf; q)n (JeiCO+¢), feiCI:I-¢); q)nl 2qn 1

x

~ (q, f, qJlbe, eJla, ex(3"(oj2 Ib 4 e4 ; q)n

x

10 Wg(abfqn-\

I(afe i¢, bfeil:l; q)nl 2

fqn, befqn-l, afqn Ie, aeil:l, ae-il:l, bei¢, be- i¢; q, q).

(Rahman [1985]) 9.19 Show that the special case ex = be = -,,(, (3 = beq~ = -0, f = t of the formula in Ex. 9.18 gives the Poisson kernel for the special Askey-Wilson

Exercises

279

polynomials Tn(X; a, b, e, bca-1Iq):

(1 - b2 e2 q2n-1)(b2 e2 q -1 ,ab,ae,q .) n (tla 2)n 2 2 2 1 2 1 (1 - b e q-1 )(q, be a- , b ca- ; q)n

00

"

f='o

x Tn(X; a, b, e, bea-1Iq)(Tn(Y; a, b, e, bea-1Iq))

= (1 _ t 2 ) (beqt; q)oo

f=

(tlbe; q)oo n=O X 10 W g(a 2q-n Ibe;

+

(q, be 2a-1, b2ea-1, be, bea- 2 , bcqt, beqlt; q)n

aq1-n Ib 2e, aq1-n Ibe 2 , q-n, aei(i, ae-i(i, aeiq" a-iq,; q, q)

2 2 q) 12 (t 'dtle " b e . q) 00 I(aei(i , beiq, , cteiq, 'bea-1tei(i. . '.00

1 2 1 2 1 belt· q)> I(te'(O+q,) , te,(q,-(i)·,ex> q) 12 ( ab 'ae " be ae- , b ea- , be a- t " CX 00

x

(beq~, -be, -beq~; q)nl(bea-1ei(i, bea-1eiq,; q)nl 2qn

~

(-t, tq~, -tq~, be2a- 1t; q)nl (tei((iH) , tei((i-q,); q)nl 2 n (q, qtlbe, qt 2 , atle; q)nl(eteiq" tbea-1ei(i; q)nl 2 q

x lOWg(bc2tqn-1 la; tqn, betqn-l, etqn la, beeiq, la, bee-iq, la, eei(i, ee-i(i; q, q)

+

(t ctla b2e2.q) I(eei(i bca-1eiq, ateiq, btei(i.q) 12 '" 00, ' . ' . ' 00 (ab, be, b2 ca- 1, be 2a- 1, abt, belt; q)oo I(te'(O+q,) , te,((i-q,); q)oo 12

x

f='o

~ (-t, tq~ , -tq~ ,abt; q)n I(tei((iH) , tei(q,-(i); q)n 12 n (q, qtlbe, qt 2etla; q)nl(ateiq" btei(i; q)nl 2 q

X 10 W

g( abtqn-1; tqn, betqn-1, atqn Ie, aei(i, ae-i(i, beiq" be-iq,; q, q),

where 0 < t < l. (Rahman [1985]; Gasper and Rahman [1986]) 9.20 Prove that 00

L

q-n 2(Pl (stei(i, ste-i(i; S2; q, -q n r 2h¢1 (sei(i It, se-i(i It; S2; q, _qn)

n=-oo

(q, q, _q 2t 2I s2, -s2 Iqt 2; q) 00

(1 _ qt2) (1 _ 2'j,t cos 0 +

x

q;;2) (_qt2, _t-2, -1, -1, -q, -q, S2, S2, S-2, qs-2; q)oo

h( cos 0; st, -st, qst, sit, -qtl s) 1

1

h(cosO; q'2, -q'2, q, -q, qtls)

x 8 W 7 (qt 2; q, stei(i, ste-i(i, qtei(i Is, qte-i(i Is; q, q), where It I < 1, 0 ~ 0 ~ n, and appropriate analytic continuations of the 2 ¢1 series are used as necessary. (Koelink and Stokman [2003])

9.21 The continuous q-Hermite polynomial Hn(cosOlq) in base q-1 becomes

280

Linear and bilinear generating functions Show that

(i)

(ii)

(Ismail and Masson [1994])

10J,81 < 1, prove that

9.22 (i) For

C' (

'-q

0.) _ (q, a 2 1,82; q)=(q,82; q2)= cos ,a (2. 2) qa ,q = x ~ (e 2i ,p,e- 2i ,p;q)= Eq(cos7jJ;,8)d7jJ.

r

2n

io

h(cosnl0/,.. Q!.eilJ Q!.e-ilJ) (3 , (3

(ii) Prove that

Eq (cos 0; a )Eq(cos 0; ,8)

= ( ,8-1 (1-n)/2. ) L -a t) ,q n ,8nqn2 /4cn (cos 0; I'lq) q ,q = n=O 1', q n _,8a- 1q(n+1)/2, -1',8a- 1q(n+1)/2 . 2] 2CP1 [ n+1 ,q, a .

( 2. 2)

= ( ~;.q 2)= X

I'q

(See Bustoz and Suslov [1998] for (i), and Ismail and Stanton [2000, (5.8)] for an equivalent form of (ii).) 9.23 Prove that

L

= (abedq-1" ae ad" a' b' a' e' a'd'· q) (bb' edt·,q)2 n (_ t) n " n n=O (q, cd, be't, b'et, bd't, b'dt; q)n(abedq-1; qhn aa'

x 8 W7 (bb' edtq2n-1; beqn, bdqn, b' e' qn, b' d' qn, btl a'; q, a'tlb)

x rn(x; a, b, e, dlq) rn(Y; a', b', e', d'lq) (bb' edt, dt I e', btei<jJ, bte-i<jJ, ctei<jJ, cte-i<jJ, b'tei () , b' te- i(}, e'tei (} , e'te- i(}; q)= (be't , b' et , bd't , b'dt , bb't " edt te i (9+<jJ) , tei(IJ-<jJ) , te i (<jJ-IJ) , te- i (9+<jJ)., q) 00 X 8W 7

(bb'tlq; be ilJ , be- ilJ , b' ei<jJ, b' e-i<jJ, btl a'; q, a'tlb)

x 8 W 7 (ee'tlq; ee ilJ , ee- i (} , e' ei<jJ, e' e-i<jJ, e'tld; q, d'tle) , where ab = a'b', cd = e'd', x = cosO, and Y = coscp. (Koelink and Van der Jeugt [1999])

Notes

281

Notes

§9.3 The special case a = ab, b = aq!, d = cq!, c ----; -c of (9.3.7), in which case the expression on the right side of (9.3.7) can be combined into a sum of SW7 series, was given by Ismail, Masson and Suslov [1997]. However, formula (3.6) in their paper does not directly lead to the generating function

f: p~a,(3)(x)tn when one sets a

=

qa/2+1/4, c

= qf3/2+1/4

and takes the limit

n=O

q ----; 1-.

JS

2 ) (x; q), Ex.9.14 A product formula for Jackson's q-Bessel function which is obtainable from this formula, is used in Rahman [2000a] to evaluate a Weber-Schafheitlin type integral for these functions. Ex.9.18 This is a q-analogue of Feldheim's [1941] bilinear generating function for the Jacobi polynomials. Ex.9.20 Rosengren [2003e] computed a more general bilinear generating function in an elementary manner. Bustoz and Suslov [1998] gave a bilateral bilinear sum that generalizes the classical Poisson kernel for the Fourier series:

f

n=-oo

tlnlein(x-y) -

1 - t2 1- 2tcos(x - y)

+ t2 '

0::; t


(l) (a;

c; _x_, -y-). x - I y-1

b, b'; c; q; x, y), then

(abx - c/q)f(q2 x , qy) + (1 - bx)f(qx, y) + (c/q - a)f(qx, qy) + (x - l)f(x, y) = 0, (aby' - c/q)f(qx, q2y) + (1 - b'y)f(x, qy)

+ (c/q - a)f(qx, qy) + (y - l)f(x, y) = O. 10.13 Show that if f(x, y)

=

(f>(2)

(a; b, b'; c, c'; q; x, y) then

ab'xf(q2 x , qy) - ~ f(q 2x, y) - axf(qx, qy) q

+ (1 + ~ 10.14 Show that if f(x " y)

=

(f>(3)

b' x ) f (qx, y)

+ (x -

1) f (x, y) = O.

(a a'·" b b'·, c·, q., x , y) then

abxf(q2 x , y) - ~ f(q 2x, qy) + ~ f(qx, qy) q

+ (1 10.15 Show that if f(x,y)

ax - bx)f(qx, y)

q

+ (x -

= (f>(4)(a,b;c,c';q;x,y)

l)f(x, y) = O.

then

abxf(q2 x , q2y) - cf(q 2x, y) - (a + b)xf(qx, qy)

+ (1 + c)f(qx, y) + (x -

l)f(x, y)

=

O.

q-series in two or more variables

300

10.16 Using (6.3.3) twice show that S(a,b,e,d,j,g;A,~)

2n(A/a,A/j,~/b,~/g;q)=

ab(l - q)2(q, q, q, cd, fla, aq/ j, g/b, qb/g, aj, bg; q)=

r (qu/a,qu/j,Au;q)= r (qv/b,qv/g,~v,eduv;q)=d b

x

if

(eu,du,Au/aj;q)=

ig

(ev,dv,uv,~v/bg;q)=

d qU

2n(abed,A/a,Aa,~/b,~b;q)=

(q, ab, ae, ad, be, bd, cd, j la, ja, g/b, gb; q)= x

+

ff

(ab;q)m+n(ae,ad,Alj;q)m(be,bd,~/g;q)n qm+n m=O n=O (abed; q)m+n(q, aq/ j, Aa; q)m(q, qb/g, ~b; q)n idem (a; j)

2n(aedg,A/a,Aa,~/g,~g;q)= +-;-----'--.,--=-'----'=-....,....:-'c-:-.::..c....:....::-'-:--=:'--:-----:--

(q, ae,ag,ad,eg,dg, cd, j/a,aj,b/g, bg; q)=

x

+

ff

(ag; q)m+n(ae, ad, AI j; q)m(eg, dg, ~/b; q)n qm+n m=O n=O (aedg; q)m+n(q, aq/ j, Aa; q)m(q, gq/b, ~g; q)n idem (a; j).

10.17 Extend (10.3.4) to

where D is a q-Lauricella function. (Andrews [1972]) 10.18 Prove that

[q-n : ale, e/d, e/e ; e/a, bid, b/e.. ] 1:2;2 be/de : a, q1-n /b ; b, q1-n /a ,q, q, q

1:3;3

(e, able; q)n (a,b;q)n ' where n = 0,1, .... Deduce the summation formula

ff

j=O k=O

(a/e, e/d: e/e; q)j(e~a, bid, b~e; qh b1akqjk (be/de, q)j+k(q, a, q)j(q, b, q)k

(e, able; q)= (a, b; q)= . (Gasper [2000])

qV,

Notes

301

10.19 Show that

j,k,m?O (,x,qz/"~y,,y,axz/,,~yz;q)oo

(x, a~zh, ~,y, y, xz, yZ; q)oo X 8 W7(~,yq-\~,

I, ,y/a, 1/z, ~Y/X; q, axzh)·

(Krattenthaler and Rosengren [2003]) 10.20 Extend (1.9.6) to

ff

n=Om=O X

(blqm 1 ,

... ,

brqmr; q)n+m(a, b; q)n(c, d; q)m

(bl , ... , br ; q)m+n(q, bq; q)n(q, dq; q)m

(a-lql-Mt(c-lql-M)m

(q, q, bq/ a, dq/ C; q)oo (bd db; q)ml ... (b r /bd; q)mr (bd)M (bq, dq, q/a, q/c; q)oo(b l ; q)ml ... (b r ; q)mr

where ml, ... , mr are nonnegative integers, M Iql-MI < min(lal, Icl)·

=

ml

+ ... +

m r , and

(Denis [1988])

Notes §1O.2 B. Srivastava [1995] introduced some elementary bibasic extensions of (10.2.5)-(10.2.8). For a quantum algebra approach to multivariable series, see Floreanini, Lapointe and Vinet [1994]. Sahai [1999] considered the q-Appell functions from the point of view of universal enveloping algebra of 8£(2). See also Jain [1980a,b] and, for connections of Appell functions with Ben root systems, Beerends [1992]. §§10.3-1O.6 Agarwal [1974] evaluated some q-integrals of the double series in (10.2.5)-(10.2.8). Also see Nassrallah [1990, 1991]. Exercises 10.1-10.2 Upadhyay [1973] gave some transformation and summation formulas for q-J ackson type double series. Ex. 10.3 This is a q-analogue of a quadratic transformation formula for Fl(a; b, b; C; x, -x) obtained by Ismail and Pitman [2000]. Exercises 10.12-10.15 Agarwal [1954] gave some 3-term contiguous relations satisfied by (1), ... ,(4), including formulas connecting them with the q-derivatives. Ex.1O.17 For an application of the q-Lauricella series, see Bressoud [1978]. Ex.1O.18 See Van der Jeugt, Pitre and Srinivasa Rao [1994] for more summation formulas of this type. For similar results when q ---+ 1, see Pitre and Van der Jeugt [1996].

11 ELLIPTIC, MODULAR, AND THETA HYPERGEOMETRIC SERIES

11.1 Introduction

Since the series L Un is called a hypergeometric series if 9 (n) = U n +IiUn is a rational function of n, and a q-(basic) hypergeometric series if g( n) is a rational function of qn, it is natural to call this series an elliptic hypergeometric series if g( n) is an elliptic (doubly periodic meromorphic) function of n with n considered as a complex variable. One motivation for considering these three classes of series is Weierstrass's theorem that a meromorphic function J(z) which satisfies an algebraic addition theorem of the form P(f(U), J(v), J(u

+ v)) = 0

identically in u and v, where P(x, y, z) is a nonzero polynomial whose coefficients are independent of u and v, is either a rational function of z, a rational function of e AZ for some A, or an elliptic function (see, e.g., Erdelyi [1953, §13.11]' Rosengren [200la, 2003c], and, for a proof, Phragmen [1885]). Elliptic analogues of very-well-poised basic hypergeometric series were introduced rather recently by Frenkel and Turaev [1997] in their work on elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation found by Baxter [1973], [1982] and Date et al. [1986-1988]. Frenkel and Turaev showed that the elliptic 6j-symbols are multiples of the very-well-poised 12Vl1 elliptic (modular) hypergeometric series defined in §11.3 (recall from §7.2 that the ordinary q-analogues of Racah's 6j-symbols are multiples of balanced 4¢3 series, which are transformable to 8¢7 series) and then used the tetrahedral symmetry of the elliptic 6j-symbols and the finite dimensionality of cusp forms to derive elliptic analogues of Bailey's transformation formula (2.9.1) for terminating 1O¢9 series and of Jackson's 8¢7 summation formula (2.6.2). This quickly led to a flurry of activity on elliptic hypergeometric series, resulting in several related papers (listed in alphabetical order) by van Diejen and Spiridonov [20002003], Gasper and Schlosser [2003], Kajihara and Noumi [2003], Kajiwara, Masuda, Noumi, Ohta and Yamada [2003], Koelink, van Norden and Rosengren [2003], Rosengren [200la-2003f], Rosengren and Schlosser [2003a,b], Spiridonov [2000-2003c], Spiridonov and Zhedanov [2000a-2003], Warnaar [2002b2003e], and others. In particular, general elliptic hypergeometric series and their extensions to theta hypergeometric series were introduced and studied by Spiridonov [2002a-2003a], who also considered theta hypergeometric integrals in Spiridonov [2003bj. Transformations of elliptic hypergeometric integrals are considered in Rains [2003bj. We start in §11.2 with the elliptic shifted factorials, Spiridonov's [2002aj "multiplicative" r+lEr theta hypergeometric series notation and its very-well302

11.2 Elliptic and theta hypergeometric series

303

poised r+1 Vr special case, and then point out some of their main properties. The "additive" forms of these series r+1er and r+1Vr are presented in §11.3, along with their modular properties, and the Frenkel and Turaev summation and transformation formulas. In §11.4 we present a simpler derivation of the elliptic analogue of Jackson's 8¢7 summation formula via mathematical induction, which is a modification of proofs that were kindly communicated to the authors by Rosengren (in a June 13, 2002, e-mail message), Spiridonov (in a Nov. 22, 2002, e-mail message) and Warnaar (see §6 in his [2002b] paper). Central to this method of proof and, in fact, to all of the formulas in this chapter is the theta function identity given in Ex. 2.16(i) which is simply the elliptic analogue of the trivial identity

(1- xA)(I- xIA)(I- fLV)(I- fLlv) - (1- xv)(I- xlv)(I- AfL)(I- fLIA)

= ~ (1 - XfL)(1 - xl fL)(1 - Av)(1 - Ajv).

(11.1.1)

The elliptic analogue of Bailey's 1O¢9 transformation formula is derived in §11.5, along with some other transformation formulas. Theta hypergeometric extensions of some of the summation and transformation formulas in sections 3.6-3.8 are derived in §11.6. Some multidimensional elliptic hypergeometric series are considered in §11. 7, where we present Rosengren's [2003c] elliptic extension of Milne's [1985a] fundamental theorem and related formulas. Many additional formulas involving theta hypergeometric series and elliptic integrals are presented in the exercises at the end of the chapter.

11.2 Elliptic and theta hypergeometric series

Define a modified Jacobi theta function by

O(X;p)

=

O(X1, ... , xm;p) =

(x,plx;p)oo,

II O(Xk;P),

(11.2.1)

k=l

where X,X1, ... ,Xm -=I- 0 and Ipi < 1. Since O(x;p) = (x;p)oo(plx;p)oo is the product of two infinite p-shifted factorials, analogous to the name of the triple product identity (1.6.1), we will call O(x;p) the double product theta function (with argument x and nome p). From (1.6.9) and (1.6.14) it follows that (11.2.2) and

[a· (J "

T]

=

O(qa. p) 'q(1-a)/2

O(q;p)

(11.2.3)

with (11.2.4) where a, (J, T are complex numbers such that 1m (T) > 0 and (J -=I- m + nT for integer values of m and n. We set qa = e2niaa and pa = e 2nim . Following

304

Elliptic, modular and theta hypergeometric series

Warnaar [2002b], we define an elliptic (or theta) shifted factorial analogue of the q-shifted factorial by

(a; q,P)n

=

{ 1,rr~:~ B(aqk;p), 1/ rr- B( aq n+k.,p ) , n

k=O-

l

and let

n = 1,2, ... , n=O,

(11.2.5)

n=-I,-2, ... , m

(aI, a2,···, am; q,P)n

=

II (ak; q,P)n,

(11.2.6)

k=l

where a, al, ... , am -I=- O. Analogous to the name q-shifted factorial for (a; q)n, we also call (a; q, P)n the q, P -shifted factorial in order to distinguish it from the a, T-shifted factorial defined in the next section. Notice that B(x; 0) = 1- x and thus (a; q, O)n = (a; q)n. Since q is called the base in (a; q)n and P is called the nome in B( a; p), we call q and P in (a; q, P)n the base and nome, respectively. Similarly, in order to distinguish the modular parameters a and T, we call a the base modular parameter and T the nome modular parameter. For the sake of simplicity, we decided not to use Spiridonov's [2002a,b] notation B(a;p; q)n for the elliptic shifted factorial. Corresponding to Spiridonov [2002a], we formally define an r+IEr theta hypergeometric series with base q and nome p by

(11.2.7) where, as elsewhere, it is assumed that the parameters are such that each term in the series is well-defined; in particular, the a's and b's are never zero. Unless stated otherwise, we assume that q and p are independent of each other, but we do not assume that the above series converges or that the numerator parameters aI, a2, ... , ar+l, denominator parameters b1, ... , bTl and the argument z in it are independent of each other or of q and p. Note that if ajpk is a nonpositive integer power of q for some integer k and a j E {I, 2, ... , r + I}, then the series in (11.2.7) terminates. Clearly, if z and the a's and b's are independent of p, then lim r+1Er(al, ... , ar+l; b1, ... , br ; q,p; z)

p--70

= r+1Er(al, ... , ar+l; b1, ... , br ; q, 0; z)

= r+lcPr(al, ... , ar+l; b1,···, br ; q, z),

(11.2.8)

where the limit of the series is a termwise limit. As is customary, the notation r+1Er(al, a2, ... , ar+l; bl , ... , br ; q,p; z) is also used to denote the sum of the series in (11.2.7) inside the circle of convergence and its analytic continuation (called a theta hypergeometric function) outside the circle of convergence. Unlike nonterminating r+1Fr series and nonterminating r+lcPr series with 0 < Iql < 1, which have radius of convergence R = 1 (as can be easily seen by applying the ratio test to (1.2.25) and (1.2.26)),

11.2 Elliptic and theta hypergeometric series

305

for 0 < Iql, Ipl < 1 and any R E [0,00] there is a nonterminating r+lEr series with radius of convergence R. In particular, there are nonterminating r+lEr series with 0 < Iql,lpl < 1 that converge to entire functions of z, which is not the case for nonterminating r+lFr and r+l¢r series with r = 0,1, ... and 0 < Iql < 1. For example, consider a nonterminating r+lEr series with 0< Iql, Ipi < 1 and (11.2.9) where ml, ... , mr+l are integers and br + l function identity ()(apm;p) (apm; q,P)n

for m, n

=

q. Since the double product theta

= (-a)-mp-(';)()(a;p)

=

implies that

(a; q,P)n( _a)-mnp-nG)q-m(~)

= 0, ±1, ... , we find that 00

. Z) -- " ( ZPr )n qMrG) , (11210) 1'+1 E l' (al,···, ar+l,. alPml , ... , arp mr . ,q,p, ~ .. n=O

with ar+l = qp-m r + 1 , Pr = rr~~i(-ak)mkp(~k), and Mr = ml + ... + mr+l. From (11.2.10) it is clear that this series converges to an entire function of Z when Mr > 0, converges only for Z = 0 when Mr < 0, and converges to 1/(1 - zPr) when Mr = 0 and IZPrl < 1. As in Spiridonov [2002a], we call a (unilateral or bilateral) series L Cn an elliptic hypergeometric series if g(n) = cn+1/cn is an elliptic function of n with n considered as a complex variable, i.e., g(x) is a doubly periodic meromorphic function of the complex variable x. For the r+lEr series in (11.2.7)

()x = II akq ;p ()(bkqX;p) 1'+1 ()(

g

x

)

Z

(11.2.11)

k=l

with br+ l = q. Clearly g(x) is a meromorphic function of x. From (11.2.4) it is obvious that qx+a- 1 = qX and hence g(x + a-I) = g(x). Since, by (11.2.1), ()(aqx+ra- 1 ;p)

=

(_aqx)-l()(aqx;p),

(11.2.12)

it follows that (11.2.13) Thus g(x

+ Ta- l ) =

g(x) when ala2··· ar+l = (bl b2 ··· br)q,

(11.2.14)

in which case g(x) is an elliptic (doubly periodic meromorphic) function of x with periods a-I and Ta- l . Therefore, we call (11.2.14) the elliptic balancing condition, and when (11.2.14) holds we say that r+lEr is elliptically balanced (E-balanced). In Spiridonov [2002a] an r+lEr series is called, simply, "balanced" when (11.2.14) holds, but here we need to distinguish between the different balancing conditions that arise. Notice that, unlike the requirement

Elliptic, modular and theta hypergeometric series

306

that z = q in the definition of a balanced r+l ¢r series, no restrictions are placed on the argument z in the above definition of an E-balanced r+IEr series. If z = q, then, by the definition of a k-balanced r+l¢r series in §1.2, the r+l¢r series in (11.2.8) is (-1 )-balanced if and only if the elliptic balancing condition (11.2.14) holds. Analogous to the basic hypergeometric special case, we call the r+IEr series in (11.2.7) well-poised if qal

=

=

a2 bl

a3 b2 =

... =

(11.2.15)

ar+lbr,

in which case the elliptic balancing condition (11.2.14) reduces to 2 2 2 ala2···ar+1

Via (11.2.5) and the n p, we find that 2

=

B(aq n;p)

alq

(11.2.16)

.

limit case of Ex. 1.1(iv) with q replaced by

---+ 00

1

()r+l

=

111

11

(qa"2,-qa"2,qa"2/p "2,-qa"2 p "2;q,P)n(_ )-n

B() a;p

1

1

1

1

1

(11.2.17)

q

1

(a"2,-a"2,a"2p"2,-a"2/p"2;q,P)n

is an elliptic analogue of the quotient 1 - aq2n

(qd, -qd; q)n

1-a

(d,-d;q)n '

which is the very-well-poised part of the r+l Wr series in (2.1.11) with al = a; see Ex. 11.3. Therefore the r+IEr series in (11.2.7) is called ve'T"y-well-poised if it is well-poised, 'T" ~ 4, and a2

=

1

qai, a3

=

1

-qai, a4

=

!

1

qai /p"2, a5

=

!

1

(11.2.18)

-qai p"2.

Corresponding to Spiridonov [2002b, (2.15)]' we define the well-poised theta hypergeometric series by

r+l Vr

very-

r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z) =

~ B(alq2n; p)

~ n=O

B(al;p )

(aI, a6, a7, ... , ar+l; q, P)n (qz)n. (q,alq / a6,alq / a7,···,alq / ar+l;q,P ) n

(

)

11.2.19

It follows that if (11.2.15) and (11.2.18) hold, then r+I Vr(al;a6,a7, ... ,ar+l;q,p;Z)

= r+IEr(al, a2,···, ar+l; bl , ... , br ; q,p; -z), and that

r+l Vr

(11.2.20)

is elliptically balanced if and only if 2 ( a62 a7···

2) 2 ar+l q =

(

alq )r-5 .

(11.2.21 )

As in Warnaar [2003c], when the argument z in the r+l Vr series equals 1 we suppress it and denote the series in (11.2.19) by the simpler notation r+l Vr(al; a6, a7, ... , ar+l; q, p). If aI, a6, a7, ... , ar+l are independent of p, then lim

p-+O

r+l Vr(al;

= r-l W r-

a6, a7,·· ., ar+l; q,p)

2(al; a6,···, ar+l; q, q),

(11.2.22)

11.2 Elliptic and theta hypergeometric series

307

which shows that there is a shift r ----t r-2 when taking the P ----t 0 limit, and that the P ----t 0 limit of an r+l Vr(al; a6, a7,"" ar+l; q,p) series with aI, a6, a7,"" ar+l independent of p is an r-l W r - 2 series. As mentioned in the Introduction, Frenkel and Thraev showed that the elliptic 6j-symbols, which are elliptic solutions of the Yang-Baxter equation (formula (1.2. b) in their paper) found by Baxter [1973], [1982] and Date et al. [1986-1988] can be expressed as 12 Vu series (in the additive notation discussed in the next section). Then they employed the tetrahedral symmetry of the elliptic 6j-symbols, which is analogous to the symmetry of the classical, quantum and trigonometric 6j-symbols (see Frenkel and Turaev [1995, 1997]), and the finite dimensionality of cusp forms (see Eichler and Zagier [1985]) to derive (in their additive form) the following elliptic analogue of Bailey's 1O¢9 transformation formula (2.9.1) 12 Vu(a;

b, c, d, e, I, Aaqn+l /el, q-n; q,p)

(aq,aq/el,Aq/e, Aq/l;q,P)n (aq/e,aq/I,Aq/el,Aq;q,P)n

x

12 VU(A;

Ab/a, Ac/a, Ad/a, e, I, Aaqn+l/el, q-n; q,p)

(11.2.23)

for n = 0,1, ... , provided that the balancing condition

(11.2.24)

bcdel(Aaqn+l/ef)q-nq = (aq)3,

which is equivalent to A = qa 2/bcd, holds. Note that both of the series in (11.2.23) are E-balanced when (11.2.24) holds. Setting A = aid in (11.2.23) yields a summation formula for 10 Vg series that is an elliptic analogue of Jackson's 8¢7 summation formula (2.6.2) and of Dougall's 7F6 summation formula (2.1.6), which, after a change in parameters, can be written in the form: TT (

lOv9

• b d -no ) _ (aq, aq/bc, aq/bd, aq/cd; q,P)n a, ,c, ,e,q ,q,p - (aq /b ,aq / c,aq /d ,aq /b c d ;q,p ) n

for n = 0,1, ... , provided that the balancing condition bcde can be written in the form

=

(

11.2.25

)

a 2qn+l, which

(11.2.26) holds. Clearly, if a, b, c, d, e are independent of p, then (11.2.25) tends to Jackson's 8¢7 summation formula (2.6.2) as p ----t O. Unlike in the basic hypergeometric limit cases of (2.6.2) discussed in Chapters 1 and 2, one cannot take termwise limits of (11.2.25) to obtain a 3E2 analogue of the q-Saalschiitz formula (1.7.2), 2El analogues of the q-Vandermonde formulas (1.5.2) and (1.5.3), or even a lEo analogue of the terminating case of the q-binomial theorem (1.3.2) in Ex. 1.3(i). Therefore one cannot derive (11.2.25) by working up from sums at the lEo, 2El, and 3E2 levels as was done in Chapters 1 and 2, and so one is forced to employ a different approach, such as in the abovementioned Frenkel and Thraev derivation, or by some other method. Rather than repeating the Frenkel and Thraev [1997] derivation of (11.2.25), we will present in §11.4 a simpler derivation of (11.2.25) via mathematical induction, which is a modification of those discovered independently by Rosengren, Spiridonov, and Warnaar. We will then show in §11.5 that (11.2.23) follows from

Elliptic, modular and theta hypergeometric series

308

(11.2.25) in the same way as in our derivation of Bailey's 1O¢9 transformation formula (2.9.1) from Jackson's 8¢7 summation formula (2.6.2). Observe that if we set e = ±(aq)~, then (11.2.25) reduces to V; ( . b

9 8

d -no ) _ (aq, aqjbc, aqjbd, aqjcdi q,P)n a, ,c, ,q ,q,p - (aq jb ,aq j c,aq jd ,aq jb cd iq,P ) n

(

11.2.27

)

for n = 0, 1, ... , provided that the balancing condition

(11.2.28) holds, which is equivalent to the elliptic balancing condition b2c2d 2 = a 3q2n+1. In view of (11.2.24), (11.2.26), (11.2.28), and of the required balancing conditions for other significant special cases of the Frenkel and Turaev transformation formula (11.2.23) to hold, analogous to the definition of a VWP-balanced series given in §2.1 we call the series r+l Vr(ali a6, a7, ... , ar+li q,Pi z) a verywell-poised-balanced (VWP-balanced) series when the very-well-poised balancing condition (11.2.29) holds. It follows that r+lVr(alia6,a7, ... ,ar+liq,P) is VWP-balanced if and only if (11.2.30) and that the summation formulas (11.2.25), (11.2.27) and the transformation formula (11.2.23) hold for n = 0,1, ... , when the series are VWP-balanced. Note that (11.2.30) reduces to (a6a7··· a2j+6)q = (alq)j when r = 2j + 5 is odd. It should also be noted that if either (11.2.30) or (a6··· ar+dq = -(±(alq)~t-5 holds, then r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, and hence an elliptic hypergeometric series. If r+l Vr(ali a6, a7, ... , ar+li q,p) is E-balanced, then it is VWP-balanced when r is even, but not necessarily when r is odd. In particular, the elliptic balancing condition (11.2.21) is not a sufficient condition for the Frenkel and Turaev transformation and summation formulas (11.2.23) and (11.2.25) to holdi the series in these formulas need to be VWP-balanced in order for these formulas to hold for n = 0,1, .... Since, if r+lEr is a well-poised series satisfying the relations in (11.2.15), r+lEr(al, a2,···, ar+li bl , ... , bri q,Pi z)

=

1.

1.

1.

1

1.

1

r+9 Vr+8 (ali at , -at, at p2, -at jp2 ,a2, a3, ... ,ar+li q,Pi -z), (11.2.31)

we find that the very-well-poised balancing condition for the above r+9 Vr+8 series is equivalent to the well-poised balancing condition (ala2··· ar+dz = - ( ± (alq) !)r+l 2

(11.2.32)

for the r+lEr series in (11.2.31). Hence, a well-poised r+lEr series is called well-poised-balanced (WP-balanced) when (11.2.32) holds. In particular, the well-poised 4E3 series in the transformation formula in Ex. 11.6 is WP-balanced. Clearly, every VWP-balanced theta hypergeometric series is WP-balanced and, by the above observations, every WP-balanced theta hypergeometric series can be rewritten to be a VWP-balanced series of the form in (11.2.31).

11.2 Elliptic and theta hypergeometric series

309

Analogous to the bilateral r'¢r series, we follow Spiridonov [2002a] in defining a rG r bilateral theta hypergeometric series by

(11.2.33) Note that rGr(al, ... ,ar;q,bl,b2, ... ,br-l;q,p;Z)

= rEr-l (al' ... ' ar; bl , ... , br- l ; q,p; z)

(11.2.34)

and, more generally, as in the bilateral basic hypergeometric case in (5.1.5), if the index of summation in an r+lEr series is shifted by an integer amount, then the resulting series is an r+l G r + l series multiplied by a quotient of products of q,p-shifted factorials. Also note that corresponding to (11.2.9) we can consider nonterminating rG r series with 0 < Iql, Ipi < 1 and

where ml, ... , mr are integers. As in (11.2.10) we find that 00

rGr(al, ... ,ar;alpml, ... ,arpmT;q,p;z)

=

L

(zur)nqNrG)

(11.2.35)

n=-oo

with U r = I1~=l (_ak)mkp("';'k) and N r = ml + .. ·+mr, which clearly converges for any z =I- 0 if and only if N r > O. If, as in Spiridonov [2002a], we replace r + 1 by r in the upper limit of the product in (11.2.11) and proceed as in the derivation of the condition (11.2.14) for an r+lEr series to be elliptically balanced, we find that the series rG r is elliptically balanced (E-balanced) if and only if ala2··· ar

= bl b2 ··· br ·

(11.2.36)

If rG r is a well-poised series with (11.2.37) then the elliptic balancing condition (11.2.36) reduces to al2 a22 ... a 2r =

( al b)r 1 .

(11.2.38)

In view of (11.2.32), (11.2.34), and (5.1.7), it is consistent to call a well-poised rG r series well-poised-balanced (WP-balanced) when (ala2··· ar)z = - (

± (albd!f.

(11.2.39)

In our consideration of modular series in the next section we are led to consider additive forms of special cases of the rather general power series

(11.2.40)

310

Elliptic, modular and theta hypergeometric series

and the Laurent series

rCis(al, ... ,ar;b1, ... ,bs;q,p;~,z) _

-

~ (a1, ... ,ar;q,P)nA n ~ nZ , n=-oo (b1, ... ,bs;q,P)n

(11.2.41)

where ~ = {An} is an arbitrary sequence of complex numbers. Some special cases of these series with An = qcm(n-l)/2 or, more generally, with An = exp(aln+a2n2 +a3n3) are considered in Spiridonov [2002a, 2003bj. If An = 1 for all n, then we will suppress ~ from the left sides of (11.2.40) and (11.2.41). When we encounter series with more than one base or nome, such as in Ex. 11.25(i), Ex. 11.26 and in several of the formulas in §11.6, or multivariable formulas, such as the multivariable extension of the Frenkel and Thraev summation formula in §11. 7, we will write the series in terms of elliptic shifted factorials. To help keep the size of this book down we will not repeat the main elliptic identities and summation and transformation formulas in the appendices. Nevertheless, for the convenience of the readers we collect below some of the most useful identities involving ()(a;p), the q,p-shifted factorials, and the q, P - binomial coefficients.

()(a;p) = ()(pja;p) = -a ()(lja;p) ()(a 2;p2) = ()(a, -a;p),

=

-a()(ap;p),

(11.2.42) (11.2.43)

()(a;p) = ()(a, ap;p2) = ()(a~, -a~, (aq)~, -(aq)~ ;p),

(11.2.44)

()(a;p)

(11.2.45)

=

()(apn;p)( -a)npG) ,

(a; q,P)n()(aqn;p) = (a; q,P)n+l = ()(a;p)(aq; q,P)n, (a; q,P)n+k = (a; q,P)n(aqn; q,ph,

(11.2.46)

(a; q,P)n = (ql-n ja; q,P)n( -atqG),

(11.2.48)

(a; q,P)n (_ ~)k (~)-nk . ) _ (a,q,Pn-k-(l_nj.) , a,q,Pk a q q

(11.2.49)

(11.2.47)

(qja;q,P)n( - ~rq-G),

(11.2.50)

q,ph(qja; q,P)n -nk (aq -no,q,p)k -_ (a; (l-kj. ) q , a,q,Pn q

(11.2.51)

(aq-n;q,P)n

=

(a; q-l,P)n = (a-\ q,P)n( _a)nq-G), 1 (_qja)n (n) (a; q, p) -n = ( aq- n ;q,p ) n ( j q a;q,p) n q 2 , (a; q,P)n+k (aq n ;q,p )k = (a; q,p)k(aqk; q,P)n = , (a;q,P)n (a;q,P)n (a; q,P)n = (ap\ q,P)n( _a)nkpn(~)qkG), (a 2; q2,p2)n = (a, -a; q,P)n, (a; q,p 2hn = (a~ , -a~, (aq) ~, -(aq) ~; q, P)n,

(11.2.52) (11.2.53) (11.2.54) (11.2.55) (11.2.56) (11.2.57)

11.2 Elliptic and theta hypergeometric series

(a; q,phn (a; q,phn (a; q,P)kn

=

= =

(a, aq; q2,P)n, (a, aq, aq2; q3,P)n, (a, aq, ... , aqk-\ qk ,P)n'

Corresponding to the q-binomial coefficient

311 (11.2.58) (11.2.59) (11.2.60)

[~] q defined in Ex. 1.2, we define

the q, P - binomial coefficient (or elliptic binomial coefficient) by

(q;q,P)n

(11.2.61 )

for k = 0, 1, ... ,n. Then

[ ~]

=

[n ~ k]

q,p

= (q-~; q,ph (_qn)kq-(~). q,p

(q, q,p)k

(11.2.62)

For complex a we can employ (11.2.62) to define

[a] k when k

= 0,1, ....

= (q- o and a Z+TZ with Z+TZ = {m+Tn: m,n E Z}. Let [n;a,T]! = [l;a,T]n and

rt

m

[aI, a2,···, am; a, T]n =

II [ak; a, T]n.

(11.3.2)

k=l

Then we can formally define the additive forms of the r+IEr and series to be the series

r+l Vr

r+l er(al,a2, ... ,ar+l;bl , ... ,br ;a,T;z) =

f

[al,a2, ... ,ar+l;a,T]nzn, n=O [1, bl , ... , br ; a, T]n

(11.3.3)

and r+I V r(al;a6,a7, ... ,ar+l;a,T;z)

=

f

n=O

[al + 2n; a, T] [al;a,T]

[aI, a6, a7,···, ar+l; a, T]n n [1,1+al-a6,1+al-a7, ... ,1+al-ar+l;a,T]n z , (11.3.4)

respectively, where, as usual, it is assumed that the parameters are such that each term in these series is well-defined. When the argument z in the r+ I vr

11.3 Additive notations and modular series

313

series equals 1 we suppress it and denote the series in (11.3.4) by the simpler notation r+lVr(al; a6, a7, ... , ar+l; a, T). Since, by (11.2.3) and (11.3.1), [a; a, T]n = (qa; q,P)n qn(b-a l /2 (qb;q,P)n [b;a,T]n

(11.3.5)

and (11.3.6) we have that r+ler(al, ... ,ar+l;b1, ... ,br;a,T;Z) . q p. zq(1+b 1 +·+br-(al +.+ar+ dl /2) = r+l E r (qa 1 , ... , qar+ 1 "• qb 1 ... ,qbr '"

(11.3.7) and

(11.3.8) From (11.2.4) and the multiplicative form of the elliptic balancing condition in (11.2.14), it follows that r+ler is an elliptic hypergeometric series when (11.3.9) which is the additive form of the elliptic balancing condition. Hence, r+ler is called E-balanced when (11.3.9) holds, and it follows that

(11.3.10) holds with the plus sign of the ± when a[al + a2 + ... + ar+l - (1 + b1 + b2 + ... + br )] is an even integer and the minus sign when it is an odd integer. Using (11.2.4) we find that the additive form of the multiplicative well-poised condition (11.2.15) is (11.3.11) in which case r+ler is called well-poised, and the elliptic balancing condition (11.3.9) reduces to (11.3.12) Note that (11.3.9) and (11.3.11) are not equivalent to the additive forms of the "balanced" and "well-poised" constraints in Spiridonov [2002a, (20) and (26)], respectively, which correspond to requiring that the expressions inside the square brackets in (11.3.9) and (11.3.11) are equal to zero.

314

Elliptic, modular and theta hypergeometric series

If r+ler is a well-poised series that can be written in the very-well-poised form (11.3.4), e.g., when, via (11.2.17) and (11.3.6), a5 =

al 1 +T 1+-+-2 2IT' (11.3.13)

then the elliptic balancing condition reduces to r - 5 ] 2IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) EZ,

(11.3.14)

which is the condition for the r+lVr series in (11.3.4) to be elliptically balanced (E-balanced) and hence an elliptic hypergeometric series. Since e27ri (1'fl)/4 = ±1, it follows by setting z = qW that the additive form of the very-well-poised balancing condition in (11.2.29) is

r-5 ] 1=f1 IT [W + 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) + -4-(r + 1) E Z, (11.3.15) which, via (11.3.8), is the condition for the series

(11.3.16) to be very-well-poised-balanced (VWP-balanced) , where, as elsewhere, either both of the upper signs or both of the lower signs in =f1 and ±1 are used simultaneously. In particular, setting w = 0, it follows that if r is odd, then r+l vr(al; a6, a7,···, ar+l; IT, T) is VWP-balanced when

r- 5 IT [ 1 + a6 + a7 + ... + ar+l - -2-(a1 + 1) ] E Z,

(11.3.17)

and if r is even, then r+lVr(al;a6,a7, ... ,ar+l;IT,T;±1) is VWP-balanced when

r-5] 1=f1 IT [ 1+a6+a7+···+ar+1--2-(al+1) +-4- EZ.

(11.3.18)

Thus, by replacing the parameters a, b, c, d, e in (11.2.25) by qa, qb, qC, qd, qe, respectively, and applying (11.3.8), we obtain the additive form of the Frenkel and Turaev summation formula:

lOV9(a; b, c, d, e, -n; IT, T) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n [a + 1 - b, a + 1 - C, a + 1 - d, a + 1 - b - C - d; IT, T]n for n

= 0, 1, ... , provided that IT[b +

C

(11.3.19)

the series is VWP-balanced, i.e.,

+ d + e - n - 2a - 1] E Z.

(11.3.20)

Similarly, it follows from (11.2.27), (11.2.28), and (11.3.8) that

9V8(a; b, c, d, -n; IT, T; ±1) [a + 1, a + 1 - b - c, a + 1 - b - d, a + 1 - c - d; IT, T]n [a + 1 - b, a + 1 - c, a + 1 - d, a + 1 - b - c - d; IT, T]n

(11.3.21 )

11.3 Additive notations and modular series

315

for n = 0,1, ... , provided that the series is VWP-balanced, i.e., 3 - -1] a [b + c + d - n - -a 2 2

1 =f-1 +4

E Z.

(11.3.22)

This formula also follows by setting e = (a+1)/2+k/2a for k = 0,1 in (11.3.19) and using the fact that [(a+1)/2+k/2a; a, T]n = (-1)k[(a+1)/2-k/2a; a, T]n. In addition, the Frenkel and Turaev transformation formula (11.2.23) has the additive form b, c, d, e, f, A + a + n + 1 - e - f, -n; a, T) + 1, a + 1 - e - f, A + 1 - e, A + 1 - f; a, T]n [a + 1 - e, a + 1- f, A + 1 - e - f, A + 1; a, T]n 12Vl1 (A; A + b - a, A + c - a, A + d - a, e, f, A + a + n + 1 -

12Vl1 (a;

[a

X

e - f, -n; a, T)

(11.3.23)

for n = 0,1, ... , provided that the series is VWP-balanced, i.e., a[b + c + d + A - 2a - 1] E Z.

(11.3.24)

The summation formula (11.3.19) follows from (11.3.23) by setting A = a-d. Frenkel and Turaev [1997] used a slightly different notation for the z = 1 case of the series in (11.3.4) (with r-IWr-2 denoting an r+IVr series when z = 1) and assumed that the series terminates (to avoid the problem of convergence) and satisfies the balancing condition r-5 1 + (a6 + a7 + ... + ar+l) = -2-(a l + 1). (11.3.25) They called the function that is the sum of their terminating series a modular hypergeometric function and justified the name "modular" by showing that if the series r+IVr(al; a6, a7,"" ar+l; a, T) terminates and (11.3.25) holds, then these functions are invariant under the natural action of 8L(2, Z) on the a and T variables, i.e., if aI, a6, a7, ... , ar+l E C, then

b)

aT + a r+IVr ( al; a6, a7,"" ar+l; - - d ' - - d CT+ CT+ = r+I Vr(al;a6,a7, ... ,ar+l;a,T)

for all

(~ ~)

E 8L(2, Z), where 8L(2, Z) is the modular group of

(11.3.26)

2 x 2 ma-

trices such that a, b, c, d E Z and ad - bc = 1. Therefore, we call (11.3.25) the modular balancing condition, and when it holds we say that the series r+IVr is M-balanced. Note that if the series r+IVr(al;a6,a7, ... ,ar+l;a,T) is M-balanced, then it is E-balanced and VWP-balanced, but if it is E-balanced or VWP-balanced then it is not necessarily M-balanced. Frenkel and Turaev also showed that if aI, a6, a7, ... , ar+l E Z, (11.3.25) holds, and the series r+l vr(al; a6, a7, ... ,ar+l; a, T) terminates, then the sum of the series is invariant under the natural action of Z2 = {(m, n): m, n E Z} on the variable a, i.e., r+l vr(al;

a6,···, ar+l; a

+ m + nT, T)

= r+I Vr(al;a6, ... ,ar+l;a,T)

(11.3.27)

316

Elliptic, modular and theta hypergeometric series

for all (m, n) E Z2, and hence is elliptic in a with periods 1 and T. A formal unilateral or bilateral series of the form Ln en (a, T) is called a modular series (or modular invariant) if it is invariant under the natural action of SL(2, Z) on the a and T variables, i.e., (11.3.28)

for all

(~ ~)

E SL(2, Z), where, as usual, two formal series are considered

to be equal if their corresponding nth terms are equal. Spiridonov [2002a] extended (11.3.26) by proving that the unilateral theta hypergeometrie series

(11.3.29) is modular if

r

= s

+ 1,

(11.3.30a) (11.3.30b)

and ai

+ a~ + ... + a;+l = 1 + bi + b~ + ... + b;,

(11.3.30e)

and, more generally, that the bilateral theta hypergeometrie series

(11.3.31 ) is modular if

r = s,

(11.3.32a) (11.3.32b)

and a 21

+ a 22 + ... + a r2 = b21 + b22 + ... + b2r'

(11.3.32e)

where it is assumed that z and the a's and b's are complex numbers such that the series are well-defined. Hence, res and rgs will be called M-balaneed when (11.3.30a)-(I1.3.30c) and (11.3.32a)-(I1.3.32c), respectively, hold. Notice that if rgr is well-poised with k

then

=

1, ...

,r,

(11.3.33)

11.3 Additive notations and modular series

317

and so this well-poised rgr series is modular when the modular balancing condition r (11.3.35) al + a2 + ... + ar = "2 (a + 1) holds. Similarly, a well-poised r+ler series with

k = 1, .. . ,r,

(11.3.36)

is modular when the modular balancing condition al

+ a2 + ... + ar+l =

r+1

-2-(a1 + 1)

(11.3.37)

holds, which reduces to (11.3.25) if (11.3.13) also holds. In order to prove the Frenkel and Turaev, and the Spiridonov modularity results, (11.3.27), and some more general results, we start by observing that, by (1.6.9) and Whittaker and Watson [1965, §21.11 and §21.5]' 'l9 1(x, e7riT) satisfies the modular symmetry relations (11.3.38) (11.3.39) where the square root of -iT is chosen so that its real part is positive, and the quasi-periodicity relations 'l9 1(X 'l9 1(x

+ 7f, e7riT ) = -'l9 1(x, e7riT ) ,

+ 7fT, e7riT ) =

_e-7riT-2ix'l91(X, e7riT ).

(11.3.40) (11.3.41)

Hence, by (1.6.14), the elliptic number [a; a, T] has the modular symmetries

[a; a, T + 1]

=

[a; a, T],

(11.3.42)

[a; a IT, -liT]

=

e7ri (a 2-1)0"2 /T [a; a, T],

(11.3.43)

and the quasi-periodicity relations

+ 1, T] = [k; a + T, T] = [k; a

(_l)k+l [k; a, T],

(11.3.44)

(-1 )k+l e7ri(1-k 2 )(20"+T) [k; a, T],

(11.3.45)

where k E Z. See, e.g., van Diejen and Spiridonov [2000, 2001b], but note that (11.3.43) corrects a typo in both of these papers. Let us first consider when the bilateral series rgs is modular. Since the modular group 8L(2, Z) is generated by the two matrices

(0 -1)

10'

(11.3.46)

which, respectively, correspond to the two transformations (a, T)

----+

(a, T + 1),

(11.3.47)

it suffices to consider when each term (11.3.48)

318

Elliptic, modular and theta hypergeometric series

ofthe series in (11.3.31) is invariant under these transformations. From (11.3.1) and (11.3.42) it is clear that [a; u, T + l]n = [a; u, T]n for nEZ, and hence n E Z.

(11.3.49)

For the second transformation in (11.3.47) observe that from (11.3.1) and (11.3.43) it follows that

nEZ, with vn(a)

(11.3.50)

= n[a 2 + a(n - 1) + (n - 1)(2n - 1)/6 - 1], and thus (11.3.51)

nEZ, with

'\n =n[ai + a~ + ... + a; - (bi + b~ + ... + b;)] + n(n - 1)[al + a2 + ... + ar - (b 1 + b2 + ... + bs )] + n(r - s)[(n - 1)(2n - 1)/6 - 1]. (11.3.52) Consequently, (11.3.53) if and only if (11.3.54) whenever cn(u, T) -I=- O. Obviously, (11.3.54) holds for n = 0 since we have '\0 = 0, and in order for it to hold for any n -I=- 0 with (u, T) replaced by

(u/(CT+d),(aT+b)/(CT+d)) for (:

~)

E

8L(2,Z), it is necessary and

sufficient that '\n = O. Setting'\l = 0 in (11.3.52) gives

a21 + a22 + ... + ar2 = b21 + b22 + ... + b2s + r - s, which, with al series

=

-1, b1

=

(11.3.55)

1, is a necessary and sufficient condition for the

to be modular when ak -I=- 0 for k (11.3.55) and

=

2, ... , r. Setting'\l

=

'\2 = 0 gives

(11.3.56) which, with al = -2, b1 = 1, are necessary and sufficient for the series

to be modular when ak -I=- 0, -1 for k = 2, ... , r. Similarly, if Nand Mare nonnegative integers, then

rgs( -N, a2, a3, ... , ar; M + 1, b2, b3, ... , bs; u, T; z) N

=2:

(11.3.57)

11.3 Additive notations and modular series

319

is modular for arbitrary values of the a's and b's if and only if

An = 0,

-M::::;n::::;N,

(11.3.58)

which, since An is a cubic polynomial in powers of n, is equivalent to (l1.3.32a)(l1.3.32c) if and only if N + M 2:: 3. In particular, when N + M < 3 the restriction r = 8 is not needed for the series in (11.3.57) to be modular. It follows from the above observations that the rgs series in (11.3.31) is modular for arbitrary values of its parameters if and only if (l1.3.32a)-(l1.3.32c) hold. Replacing 8 by 8 + 1 in rgs and then setting bS +1 = 1 yields that the res series in (11.3.29) is modular for arbitrary values of its parameters if and only if (l1.3.30a)-(l1.3.30c) hold. It should be noted that, since the modularity of the above res and rgs series hold termwise under the stated conditions, when we extend res and rgs to

(11.3.59) and

(11.3.60) respectively, where A = {An} is an arbitrary sequence of complex numbers and aI, a2, ... , ar+l, Z E "q/e;q)n (>"q,d/a,aq/b,aq/e;q)n x 10 VVg (>..; a/ >.., J1" -J1" b, e, dqn, q-n; q, _q>..2 / a) and that the 12 VVl1 series is VWP-balanced and balanced, while the series is VWP-balanced but not balanced.

10 VVg

(See Warnaar [2002b] for part (i), and Rahman and Verma [1993] for the transformation in part (ii). Also see Andrews and Berkovich [2002] and Warnaar [2003c,e].) 11.25 (i) Extend the quadbasic transformation formula in Ex. 3.21 to

t

()(arkqk, brkq-k;p) (a, b; r,p)de, a/be; q,ph k=O ()(a,b;p) (q,aq/b; q,p)k(ar/e, ber;r,p)k (CR-n /A, R-n / BC; R, p)k(Q-n, BQ-n /A; Q, P)k k X (Q-n/C,BCQ-n/A;Q,Ph(R-n/A,R-n/B;R,Ph q (ar, br; r,P)n(eq, aq/be; q,P)n(Q, AQ/ B; Q, P)n(AR/C, BCR; R, P)n (q, aq/b; q,P)n(are, be/r; r,P)n(AR, BR; R, P)n(CQ, AQ/ BC; Q, P)n

x

t

()(ARkQk,BRkQ-k;p) (A,B;R,Ph(C,A/BC;Q,P)k ()(A, B; P) (Q, AQ/ B; Q, P)dAR/C, BCR; R, P)k k=O x (er-n la, r- n /be; r,p)k(q-n, bq-n fa; q,P)k Qk (q-n /e, beq-n fa; q,p)k(r- n la, r- n /b; r,p)k for n = 0,1, .... (ii) Deduce the following transformation formula for a "split-poised" theta hypergeometric series

t

()(aq2k;p) (a,b,e,a/be;q,p)k k=O ()(a;p) (q, aq/b, aq/e, beq; q,p)k (q-n, B/Aqn, C/Aqn, I/BCqn; q,ph k (I/Aqn, I/Bqn, l/cqn, BC/Aqn; q,P)k q (aq, bq, eq, aq/be, Aq/ B, Aq/C, BCq; q,P)n (Aq, Bq, Cq, Aq/ BC, aq/b, aq/e, beq; q,P)n

X

x

X

t

()(Aq2k;p) (A,B,C,A/BC;q,p)k k=O ()(A;p) (q, Aq/ B, Aq/C, BCq; q,ph (q-n,b/aqn,e/aqn,l/beqn;q,p)k k q (l/aqn,l/bqn, l/eqn, be/aqn; q,p)k

for n = 0,1, ... , which is an extension of the transformation formula for a split-poised 1O¢9 series given in Ex. 3.21. Write this formula as a transformation formula for a split-poised 12En series.

Exercises

343

(Gasper and Schlosser [2003])

11.26 Extend Ex. l1.25(i) to

t

()(a(rst/q)k, brkq-k, cskq-k, at k /bcqk;p) k=O ()(a, b, e, a/bc;p) (a; rst/q2,p)db; r,p)k(c; s,ph(a/bc; t,P)k (q;q,p)k(ast/bq;st/q,p)k(art/cq;rt/q,p)k(bers/q;rs/q,p)k (Q-n; Q, Ph(B(Q/ST)n /A; ST/Q, P)k(C(Q/ RT)n /A; RT/Q, P)k x ((Q2 /RST)n/A; RST/Q2, p)k(R-n / B; R, p)k(s-n /C; S, Ph ((Q/Rs)n/BC;RS/Q,P)k k x (BC/ATn;T,p)k q (arst/q2;rst/q2,P)n(br;r,P)n(cs;s,P)n(at/bc;t,P)n (q;q,P)n(ast/bq;st/q,P)n(art/cq;rt/q,P)n(bers/q;rs/q,p)n (Q; Q,P)n(AST/BQ; ST/Q,P)n X..,....,...'--'--"":""'--:'-""""'-------'--::--'-:-"""":--'----'--....,..(ARST /Q2; RST/Q 2, P)n(BR; R, P)n (ART /CQ; RT /Q, P)n(BCRS/Q; RS/Q, P)n x-'------'-----;-==-=---=-~--;--;-::::-:-=-=:--::::-_=_=_----'-....::...:....--'-(CS; S,P)n(AT/BC;T, P)n x ()(A(RST/Q)k, BRkQ-k, CSkQ-k, ATk/BCQk;p) ()(A, B, c, A/ BC; P) k=O (A; RST/Q2, P)k(B; R, P)k x~'--='--=-:-~-===-:-:'::-'-:'---,==-':-::-'--'::-:­ (Q; Q, P)k(AST/BQ; ST/Q, P)k (C; S, P)k(A/BC; T, P)k x (ART/CQ; RT/Q, P)k(BCRS/Q; RS/Q, P)k (q-n; q,p)k(b(q/ st)n fa; st/q,p)k x-'-=--------;.-;-:~-"--'-:-=:-----'-----'--;--';::_____'_,-=-:-=---'-'-'­ ((q2 /rst)n fa; rst/q2 ,ph x (c(q/rt)n fa; rt/q,ph((q/rs)n /bc; rs/q,ph Qk (r- n /b; r,p)k(S-n /e; s,p)k(bc/atn ; t,p)k x~~~~~~~~~~~~~~~~~~

t

for n = 0,1, .... Use (11.6.9) and (11.6.18) to extend this formula to a transformation formula containing the two additional parameters d and D. (Gasper and Schlosser [2003]) 11.27 Show that if

344

Elliptic, modular and theta hypergeometric series and a~ = bkckdkekfkgk for k = 0, ±1, ±2, ... , then

~ ()(aVbkckdkh, aVbkckdkek, ak, ak/ekfk;P) U k

f:::n ()(ak/ek, ak/ fk' aVbkCkdkekfk, aVbkCkdk;p) x =

[

()(ak/ckdk,ak/bkdk,ak/bkck,ek,fk,gk;p)

1

- ()(ak/bk,ak/ck,ak/dk,a~/bkckdkek,a~/bkckdkfk,a~/bkckdkgk;P) Urn - Un+1

for n,m

=

]

0, ±1, ±2, ....

(Gasper and Schlosser [2003]) 11.28 Extend the indefinite multibasic theta hypergeometric summation formula in (11.6.9) to

k a (V ) k a 2 ( v 2 ) k

a2 (V2 ) k a2

)

~ () (av 'J[j Ui 'TiCrJ! qrsl 'vr;cag qrsu 'DCa; P ~ ( a a2 a 2 a 2 ( v 2 ) k. ) () a, J[j' bcd] , bcdg' bcd qr s

k=O

,p

3

3

(b; q,p)k(C; r,ph(d; s, p)k(f; t, p)k(g; u, p)k (bcdfg; q;!stu ,p) k (qrstu/v 2)k x

--------------------------------~~~~--~--~----

a v a v a v av v av v bcdfgqrstu qrstu (-b;-,P)k(-;-,P)k(-d;-,P)k(ft;-t,P)k(-;-,P)k( 22 ; - 2 ,P)k c r s gu u a v v q

1

()(cdU!s)k, bd(~s)k, bc(~r)k,ftk,guk, bCdfg(q;!s3tu )k;p) x [ 1------------------------.;-------;o;-------,;---~---=-------;;-------a (V)k a (V)k a (V)k a 2 ( V2 )k a 2 ( v 2 )k fg (tU)k. ) () ( 7i q 'c r 'd S 'TiCrJ! qr st 'Ocag qr su 'a v ' P ()(a/f,a/g,a 2/bcdfg,a 2/bcd;p) ()(a 2/bcdg,a 2/bcdf,a,a/fg;p) x

[1 _

(b; q,P)n+l(C; r,P)n+l(d; S,P)n+l(f; t,P)n+l(g; U,P)n+l

(~; ~'P)k+l (%; ¥,P)n+l (a; ~,P)n+l Uj-; ¥,P)n+l (~; *,P)n+l a3

1

V3

x (-oCilfii; 7jFSfU' p) n+ 1 , ( bcdtg. qrstu) a

'

----;;r-' P

n+l

where n = 0,1, .... (Gasper and Schlosser [2003]) 11.29 Define the elliptic beta function BE(t;q,p) via its elliptic beta contour integral representation

BE(t; q,p)

=

rllE(Z;

IJJ'

t; q,p) dz,

z

where l' is the positively oriented unit circle, 4

I1

r(ztk,tk/Z;q,p) k=O llE z;t;q,p = -2 .r( Z, 2 1/ 2 7rZ z, zA, A/ z; q,p)' (

)

1

345

Exercises 4

t =

(to,t1,t2,t3,t4), max(ltol, It11, It21, It31, It41, Iql, Ipl) < 1, A

I1

=

tk,

k=l

Iqpl < A, and n

r(Zl,"" Zn; q,p)

=

II r(Zk; q,p),

n

=

1,2, ... ,

k=l

with f(z; q,p) as defined in Ex. 11.12. Prove that 2

BE(t; q,p)

=

I1 O~j 1, C. R. Math. Rep. Acad. Sci. Canada 15, 7-12. Ismail, M. E. H. and Masson, D. R. (1994). q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals, Trans. Amer. Math. Soc. 346, 63-116. Ismail, M. E. H. and Masson, D. R. (1995). Generalized orthogonality and continued fractions, J. Approx. Theory 83, 1-40. Ismail, M. E. H. and Masson, D. R. (1999). Some continued fractions related to elliptic functions, Contemp. Math. 236, 149-166. Ismail, M. E. H., Masson, D. R. and Rahman, M. (1991). Complex weight functions for classical orthogonal polynomials, Canad. J. Math. 43, 1294-1308. Ismail, M. E. H., Masson, D. R. and Suslov, S. K. (1997). Some generating functions for q-polynomials, Fields Inst. Commun. 14,91-108. Ismail, M. E. H., Masson, D. R. and Suslov, S. K. (1999). The q-Bessel function on a q-quadratic grid, CRM Proc. Lecture Notes 22, 183-200.

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50,

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l

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1, SIAM

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Symbol index

(a)n (a; q)n (a; q)l/ (al' a2, ... ,am; q)n (a; q)oo (al' a2, ... ,am; q)oo (a; q,P)n, (al' a2,···, am; q,P)n (a;p, q)r,s ,al, laJ [a]q [n]q! [a]q,n [al' a2,·· . ,am]q,n [a; 0"] [n; O"]! [a;O"]n [a;O",T] [n;O",T]! [a; 0", T]n, [al' a2,··· ,am; 0", T]n [al' a2,·· . ,am; 0", T]n an(a, ;3lq) A(z) bk,j b(k, n;;3) bn (a,;3; q) B(z) B((}) B(x,y) Bq(x, y) B~(x; k) BE(t; q,p) cosq(x) Ck,n

2, 5 3, 6 208 6 6 6 304 112 205 7 7 7 7 7 8 8 17 312 312 312 192 185 195 185 192 189 265 23 23 205 344 28 195

Cj (;3lq) cn(x; a; q) c~(x; k)

C

CN (Contour) Cj,k,m,n (Contour) C(¢) Cosq(x) Cq(x;w) Cn(x; ;3lq) C~(x; ;3lq) C~(x)

C.T.

'Dq Dq Dn(x; ;3lq) r+ler res eq(z) eXPq(z) Eq(z) E± q r+1Er rEs Eq(x; a) Eq(x; a, b) Eq(x, y; a) F(a, b; c; z) rFs Fl ,F2,F3 F4 F(x, t) F(x, ylq) Fl (x, ylq), F2(x, ylq) 415

187 202 204 312 126 345 265 28 212 31 255 2 171 27 197 212 312 316, 319 11 12 11 197 304 309 12 12 111

2 3 283 219, 283 259 265 266

416 FA:B;C D:E;F

r9s rGr rGs 9o(a,b,e,d,f) G(x, t) Gl(x, ylq) G2(x, ylq) Gt(x; a, b, e, dlq) Gt(x; a, elq) h(x; a), h(x; a; q) h(x; aI, a2,··· ,am) h(x; aI, a2,··· ,am; q) hn(q),hn(a,b,e,N;q) ht(a) hn(a, b, e; q) hn(a, b, e, dlq) hn(a, b, e, d, 1) hn ((3lq) hn(x; q) Hn(x) Hn(xlq) Hn(x;q) Ht(x), Ht(x; a, b, e, dlq) H(x,y, t) idem (b; e) idem (al;a2, ... ,ar+1) I(a,b,e,d) 1m Ja(x) JP) (x; q), JS 2)(X; q), JS 3)(x; q) J(a,b,e,d,j,9) kn ]( (Contour)

]((x, y, z) ]((x, y, Z; (3lq) ](n(x; a, N; q) ](n(x; a, Nlq) ](t(x, y) ](t(x, y; (3lq) ](:ff (x; a, N; q) Lm(a; b, e, d) Lt(x, y; (3lq)

Symbol index

283 316,319 309 310 159 260 269 270 262 264 154 154 154 180 261 182 190 256 186 209 4 31 209 263 259 69 121 156 125 4 30 157 265 125 225 223 201 201 227, 261 227 202 164 227

231 4 L~(x) 210 L~(x; q) Mn(x; a, e; q) 202 M(x, y), M(x, y; a, b, e, e' ; q) 291 p(n) 239 240 PN (n), Pe (n), Pdist (n), Po( n) 241 Peven (n), Podd (n) 175 Pn(x) 195 Pn(x; a, (3) 32, 181 Pn(x; a, b; q) 194 Pn(eos(B + ¢); a, blq) 59, 189 Pn(x; a, b, e, dlq) 253 Pv(qX; a, b; q) P(z) 125 2 Pn(x) Pn(x; a, b, e; q) 182 255 Pn(xlq) 255 Pn(x; a, b, e, d, a2, a3, . .. , aslq)

Lt(x, y; a, b, e, a", M, N; q)

p~a,(3)(x)

p~a,(3) (x;

2

q)

191

p~a,(3) (xlq)

191 229 229 180 273 189 243 265 254 333 181 257 345 216 28 28 315 294 61 214 257

Pz(x, y) Pz(x,y;a,b,e,a",](,M,N;q) Qn(x), Qn(x; a, b, N; q) Qn(x; a, blq) Qn(z; a, b, e, dlq) r2k(n) 224, rn(x;a,b,e,dlq),rn(x) r~(x), r~(x; a, b, e, dlq) RN Rn(~(x)), Rn(~(x); b, e, N; q) Rm(x), Rm(x; a, b, e, d, 1) Rm,j(z) sn(x) sinq(x) Sinq(x) SL(2, 2':) S(a,b,e,d,j,9;A,~) S(A,~,V,p)

Sn(x;p, q) Sn(x),Sn(x;a,b,e,d,j)

Symbol index

Sq(x; w) rts ']['

,][,n Tn (X) Tn,k(Z) Un (X) U~a)(x; q) Vn r+lVr v(x; a, b, c, d, f) V(e iO ) r+1 v;.

Wj w(x) w(x; a, b, c, d) w(x; a, b, c, dlq) w( 0; q) wk(a, b, c, dlq) W(O) W(e iO ), W(eiO,a,b,c,dlq) W,a(xlq) Wn(x; b,q) Wn(x;q), Wn(x;a,b,c,fV;q) r+lWr x+ z,zqk Z 2Z Z+rZ J J(t) dqt J(n) "-' g(n)

L:~=m ak rr~=m ak

[~L' [~L [kl' ~ ,kmL

212 7 344 347 2 345 2 209 176 312 159 198 306 175, 179 175 157 190 192 191 194 198 185 214 59, 180 39 114 331 312 320 312 23 185 81 325 24 25

0

[~Lp' [~Lp

311

[~Lp

417

312

175 21 20, 29 rq(X) r(z; q,p) 311,338 345 r(Zl'oo 0 ,zn;q,p) t(z; q,p) 338 42 Om,n 11, 197 Oq 11 /).J(z), '\l J(z) /).Uk 80 32 /).bJ(Z) /).(Z) , /).(zqk) 331 /).(z;p), /).n(z;p) 333 /).E(Z; t; q,p) 344 /).E(Z; t) 345 303 O(x;p), O(Xl' 0 0 0, xm;P) 16 171 (x, q), 172 (X, q), 173 (X, q), 174 (x, q) 190 K(a, b, c, dlq) 180 Aj 156 An(q) 156 An(a, b, c, d, a2, a3, 0 0 0, aslq) 181 JL(x) 269 Vn 270 1rn p(x;q),p(x;a,b,c,fV;q) 180 194 Pn(a, blq) 156 p(xlq) 156 p(x; a, b, c, d, a2, a3, 0 0 0, aslq) ¢(a,b;c;q,z) 3 2¢1 (a, b; c; q, z) 3 4 r¢s [0 0 oj 95 (l) , (2) ,(3), (4) 283 294 1(a; {3, (3'; 'I; q; x, y) D(a; bl , 0 0 0, br ; c; q; Xl, 0 0 0, x r ) 300 A:B;C 283 D:E;F 137 r1/Js 315 r-l W r-2 a(x) r(x)

Author index

Adams, C. R., 36 Adiga, C., 35, 67, 152 Agarwal, A. K., 111, 257 Agarwal, N., 112 Agarwal, R. P., 34-36, 68, 111, 112, 119, 122, 124, 136, 301 Aigner, M., 35 Alder, H. L., 257 Alexanderson, G. L., 35 Alladi, K., 35, 67 Allaway, Wm. R., 204, 205, 214, 216 AI-Salam, W. A., 35, 36, 52, 83, 86, 204, 205, 209, 214-216, 273 Andrews, G. E., 9, 15, 21, 26, 29, 33-37, 52, 53, 59, 65, 67, 68, 111, 112, 138, 141, 147, 149, 152, 174, 181, 182, 184, 213, 215, 216, 241-243, 245, 257, 296, 297, 300, 341, 342 Aomoto, K., 174 Appell, P., xv, 282 Artin, E., 36 Askey, R., 2, 9, 17, 21, 26, 29, 31, 33-35, 52, 53, 59, 67, 84, 99, 113, 125, 129, 136, 138, 141, 149, 152, 154, 165, 170-172, 174, 177, 180-182, 185, 188, 191, 193, 195, 197, 198, 200, 204-206, 214, 215, 232, 236, 242, 257, 273, 274, 281 Atakishiyev, M. N., 215, 257 Atakishiyev, N. M., 12, 35, 195, 213-215, 257 Atkin, A. O. L., 152 Atkinson, F. V., 176 Bailey, W. N., xxiii, 5, 9, 18, 41, 46, 47, 50, 53, 54, 57, 58, 60, 61, 64, 67, 73, 96100, 112, 140, 148-150, 152, 219, 236, 261, 287-290, 293, 298, 299 Baker, M., 35 Baker, T. H., 68 Bannai, E., 213 Barnes, E. W., 113, 117

418

Bateman, H., 221, 222 Baxter, R. J., 35, 67, 302, 307 Beckmann, P., 257 Beerends, R. J., 301 Bellman, R., 35 Bender, E. A., 34 Berg, C., 214 Berkovich, A., 35, 67, 341, 342 Berman, G., 35 Berndt, B. C., 35, 67, 152, 257 Bhargava, S., 35, 67, 152 Bhatnagar, G., 257, 258, 325, 349, 350 Biedenharn, L. C., 213 Bohr, H., 29 Baing, H., 34 Borwein, J. M., 67 Borwein, P. B., 67 Bowman, D., 36 de Branges, L., xv, 84, 232, 237, 257 Bressoud, D. M., 35, 36, 67, 68, 83, 111, 112, 213, 226, 257, 301 Bromwich, T. J. I'A., 5 Brown, B. M., 214, 216 Burchnall, J. L., 111, 112, 291 Burge, W. H., 67 Bustoz, J., 36, 213, 273, 280, 281 Carlitz, L., 28, 31, 35, 36, 58, 64, 111, 152, 209, 210, 215, 216, 275, 281 Carlson, B. C., 34 Carmichael, R. D., 36 Carnovale, G., 35 Cauchy, A.-L., 9, 112, 132 Cayley, A., 112 Charris, J., 215 Chaundy, T. W., 111, 291 Cheema, M. S., 35 Chen, Y., 216 Cherednik, 1., 68 Chihara, L., 213-215

Author index

Chihara, T. S., 34, 176, 209, 213-216, 259, 273 Chu Shih-Chieh, 2 Chu, VV., 111, 151, 152, 257, 325, 349, 350 Chudnovsky, D. V., xv, 232 Chudnovsky, G. V., xv, 232 Chung, VV. S., 258 Ciccoli, N., 215 Cigler, J., 33, 35, 112, 216 Clausen, T., xiv, 103, 232 Cohen, H., 258 Comtet, L., 258 Coon, D., 35 Cooper, S., 68, 258 Crippa, D., 34 Date, E., 302, 307 Daum, J. A., 18 Dehesa, J. S., 216 Delsarte, P., 181, 202, 213, 215 Denis, R Y., 301, 350 Desarmenien, J., 112, 216 Dickson, 1. E., 243 van Diejen, J.F., 258, 302, 317, 347, 349 Di Vizio, L., 36 Dixon, A. C., 38 Dobbie, J. M., 67 Dougall, J., 38 Dowling, T. A., 35 Dunkl, C. F., 35, 181, 202, 259 Duren, P. L., 257 Dyson, F. J., 67, 68, 242, 257 Edwards, D., 112 Eichler, M., 307 Erdelyi, A., 34, 66, 111, 174, 236, 249, 265, 284, 287, 302 Euler, 1., 3, 62, 239, 241 Evans, R J., 68, 174 Evans, VV. D., 214 Ewell, J. A., 35 Exton, H., 36, 283 Faddeev, L. D., 36 Fairlie, D. B., 34 Favard, J., 176 Feinsilver, P., 33 Felder, G., 312, 338, 339, 350 Feldheim, E., 281 Fields, J. L., 84, 86 Fine, N. J., 67, 112, 242 Floreanini, R, 215, 301

419

Floris, P. G. A., 258 Foata, D., 36, 112 Foda, 0., 67 Forrester, P. J., 67, 68, 257 Fox, C., 19 Frenkel, I. B., 7, 17, 258, 302, 307, 315 Freud, G., 176 Fryer, K. D., 35 Fiirlinger, J., 112 Gangolli, R., 195 Garoufalidis, S., 34 Garrett, K., 67, 148 Garsia, A. M., 67, 112 Gartley, M. G., 12 Garvan, F. G., 34, 35, 67, 174 Gasper, G., 19, 32-34, 36, 58, 61, 65-67, 74,78,80,81,83-85,88,91,93,94,105107, 109-111, 130, 135, 136, 172, 174, 186, 195, 207, 208, 213, 215, 217, 223, 226, 227, 229, 232, 233, 235, 236, 238, 239, 248, 250, 251, 256-258, 279, 300, 302, 326, 327, 329, 343, 344, 349, 350 Gauss, C. F., xiv, 1 Gegenbauer, 1., 225, 249 Geronimo, J. S., 258 Gessel, I., 83, 93, 112, 258, 349, 350 Goethals, J. M., 181, 215 Goldman, J., 35, 258 Gonnet, G. H., 34, 174 Gordon, B., 152 Gosper, R. VVm., 21, 34, 36, 81, 93, 111, 148 Goulden, I. P., 37, 68, 257 Greiner, P. C., 19 Grosswald, E., 243 Gupta, D. P., 215 Gustafson, R A., 35, 152, 173, 258, 349, 350 Habsieger, L., 174 Hahn, VV., 28, 30, 36, 138, 180, 181 Hall, N. A., 72 Handa, B. R., 35 Hardy, G. H., 44, 45, 138, 240-242 Heine, E., xviii, 3, 9, 13, 14, 21, 26-28 Henrici, P., 32, 34 Hickerson, D., 242, 257 Hilbert, D., 243 Hirschhorn, M. D., 152, 245, 258 Hodges, J., 215 Hofbauer, J., 112

420

Author index

Hou, Q.-H., 216 Hua, L. K., 243 Ihrig, E., 35 Ismail, M. E. H., 12, 22, 30, 31, 35-37, 61, 65, 67, 68, 86, 111, 138, 141, 148, 152, 155, 172-174, 177, 184, 185, 204, 209, 212-216, 251, 254, 257-259, 261, 272277, 280, 281, 301 Ito, M., 174,349 Ito, T., 213 Jackson, D. M., 37 Jackson, F. H., 14, 18, 21-23 28 30 32 35, 36, 43, 58, 87, 103, 11~, 1'12, '136: 138, 232, 282-284, 311, 338 Jackson, M., 144, 146, 152 Jacobi, C. G. J., 15, 34 Jain, V. K., 36, 60, 61, 63, 78, 97, 98, 101, 111, 112, 152, 232, 242, 301 Jimbo, M., 35, 302, 307 Joichi, J. T., 34, 35 Joshi, C. M., 68

Lakin, A., 141 Lam, H. Y., 258 Lapointe, L., 301 Lascoux, A., 216 Lassalle, M., 174, 201 Le, T. T., 34 Leininger, V., 258, 349 Leonard, D. A., 213 Lepowsky, J., 35, 68 LeTourneux, J., 215 Lewis, R. P., 35 Li, X., 215 Libis, C. A., 215 Lilly, G. M., 258, 349 Littlewood, J. E., 118 Liu, Zhi-Guo, 258 Lorch, L., 22 Louck, J. D., 213 Lubinsky, D. S., 216 Luke, Y. L., 34, 88

Kac, V. G., 35, 152 Kadell, K W. J., 36, 68, 152, 174, 258 Kairies, H.-H., 36 Kajihara, Y., 302, 346 Kajiwara, K, 302 Kalnins, E. G., 135, 214, 257, 258 Kampe de Feriet, J., xv, 282 van Kampen, N. G., 35 Kaneko, J., 68, 174 Karlsson, Per W., 18 Kashaev, R. M., 36 Kendall, M. G., 35 Kirillov, A. N., 258 Klein, F., xv Klimyk, A. U., 215, 257 Knopfmacher, A., 34 Knuth, D., 35, 36 Koekoek, R., 259, 273 Koelink, H. T., 213-215, 252, 256, 258, 274, 279, 280, 302 Koepf, W., 34 Koornwinder, T. H., 21, 33-35, 101, 172, 202, 210, 213-216, 257, 258, 349 Krattenthaler, C., 34, 83, 111, 112, 258, 301, 346, 349 Kummer, E. E., xv Kuniba, A., 302, 307

Macdonald, 1. G., 35, 68, 174 MacMahon, P. A., 35 Masson, D. R., 36, 173, 213, 215, 275, 276, 280, 281 Masuda, T., 37, 302 McCoy, B. M., 67, 68 Meijer, H. G., 36 Menon, P. K., 35 Merkes, E., 36 Milin, 1. M., 257 Miller, W., 34, 36, 135, 214, 258 Milne, S. C., 35, 67, 152, 174, 213, 257, 258, 303, 325, 331, 333, 335, 349, 350 Mimachi, K., 36 Minton, B. M., 18, 19 Misra, K. C., 68 Miwa, T., 302, 307 Moak, D. S., 21, 29, 211, 216 Mohanty, S. G., 35 Molk, J., 151, 331 Mollerop, J., 29 Monsalve, S., 215 Mordell, L. J., 258 Morris, W., 68, 174 Mu, Y.-P., 216 Muldoon, M. E., 22, 36 Mulla, F. S., 257 Mullin, R., 258 Muttalib, K A., 216

Laine, T. P., 216

Nakagami, Y., 37

Author index

Narukawa, A., 350 Nassrallah, B., 96, 104, 105, 112, 158, 163, 232, 295, 301, 341 Needham, J., 2 Nelson, C. A., 12 Nevai, P. G., 186, 190, 215, 259 Newcomb, J. W., 258, 350 Nikiforov, A. F., 34, 135, 136, 213 Nishizawa, M., 350 van Norden, Y., 302 Noumi, M., 258, 302, 346 O'Hara, K. M., 35 Ohta, Y., 302 Okado, M., 302, 307 Onofri, E., 257 Opdam, E. M., 68 Orr, W. McF., 112 Pastro, P. I., 216 Paule, P., 34, 67, 68, 112 Pearce, P. A., 35, 67 Perline, R, 37 Perron, 0., 176 Petkovsek, M., 34 Pfaff, J. F., xiv, 17 Phragmen, E., 302 Pitman, J., 301 Pitre, S. N., 301 P6lya, G., 35 Potter, H. S. A., 33 Qazi, Tariq M., 214, 254, 255, 257 Quano, Y.-H., 67 Rademacher, H., 241 Rahman, M., 34, 36, 58, 61, 65, 74, 78, 81, 91-94, 96, 104, 108-112, 136, 152, 153, 155, 158, 159, 161, 163, 169, 171-174, 185, 191, 207, 208, 211-217, 223, 225229, 232, 235, 248, 249, 251, 253-258, 269, 274-279, 281, 295, 341, 342, 350 Rains, E., 174, 302, 347, 349 Rainville, E. D., 34 Rakha, M. A., 258, 349 Ramanujan, S., 68, 138, 152, 172, 232, 236, 240 Ramanathan, K. G., 67 Rankin, R A., 67 Remmel, J., 112 Reshetikhin, N. Yu, 258 Riese, A., 34

421

Rogers, L. J., xviii, 31, 68, 184, 226 Rogov, V.-B. K., 35 Rosengren, H., 258, 281, 301-303, 331, 332, 335, 337, 346-350 Rota, G.-C., 35, 258 Rothe, H. A., 9 Roy, R, 9, 113, 125, 129, 172 Ruedemann, R. W., 213 Ruijsenaars, S. N. M., 311, 338 Saalschiitz, L., 17, 113 Saff, E. B., 216 Sahai, V., 301 Sauloy, J., 36 Schempp, W., 257 Schilling, A., 67, 68 Schlosser, M., 152, 174, 257, 258, 302, 326, 327, 329, 343, 344, 347, 349, 350 Schur, I. J., 68 Schiitzenberger, M.-P., 33 Schwartz, A. L., 257 Schweins, Ferd. F., 9 Sears, D. B., 15, 49, 51, 61, 64, 70, 71, 74, 130, 131, 136, 152 Selberg, A., 174 Shohat, J., 214 Sills, A. V., 34, 68 Simon, K., 34 Singh, V. N., 99, 105, 232 Slater, L. J., 5, 9, 32, 36, 68, 125, 126, 128, 130, 141-144, 151, 152, 242, 331 Spiridonov, V. P., 215, 302-306, 309, 310, 313, 316, 317, 320, 338, 341, 345, 347, 349 Srinivasa Rao, K., 301 Srivastava, B., 301 Srivastava, H. M., 84, 107, 232, 283 Stanton, D., 34-36, 65, 67, 68, 83, 93, 111, 112, 148, 155, 174, 181, 201, 213, 215, 216, 258, 276, 277, 280, 349, 350 Starcher, G. W., 36 Stembridge, J. R., 68 Stevens, L., 350 Stokman, J. V., 174, 215, 253, 258, 279 Stone, M. H., 176 Stuart, A., 35 Styer, D., 36 Subbarao, M. V., 152, 349 Sudler, C., 35 Suslov, S. K., 12, 34-36, 111, 135, 136, 152, 153, 195, 213, 214, 216, 257, 258, 274-276, 280, 281

422

Author index

Swarttouw, R. F., 36, 258, 259, 273 Swinnerton-Dyer, P., 152 Sylvester, J. J., 35 Szego, G., 31, 34, 35, 176, 177, 213, 216, 249, 259, 260, 265 Takacs, L., 2 Tamarkin, T., 214 Tannery, J., 151, 331 Tarasov, V., 174 Temme, N., 259 Terwilliger, P., 213 Thomae, J., xviii, 21, 23, 24, 69, 70 Toeplitz, 0., 35 Touhami, N., 258 Tratnik, M. V., 213, 215, 258 Trebels, W., 257 Trjitzinsky, W. J., 36 Trutt, D., 257 Turaev, V. G., 7, 17, 258, 302, 307, 315 Ueno, K., 37 Upadhyay, M., 301 Uvarov, V. B., 34, 136, 213 Valent, G., 213 Vandermonde, A. T., 2 Van Assche, W., 258 Van der Jeugt, J., 252, 258, 280, 301 Varchenko, A., 174, 312, 339, 350 Verma, A., 36, 52, 61, 63, 65, 68, 78, 83, 86, 97, 98, 111, 112, 185, 207, 214, 216, 225, 228, 242, 249, 257, 342, 349 Vidyasagar, M., 152 Viennot, G., 67, 216

Vilenkin, N. Ja., 34 Vinet, L., 215, 301 Volkov, A. Yu., 36 Wallisser, R., 36 Warnaar, S. 0., 34, 67, 68, 302-304, 306, 323, 325, 329, 339-342, 346, 347, 349 Watson, G. N., xv, 16, 17, 21, 34, 35, 42, 67, 68, 112, 114, 115, 117, 119, 124, 151, 152, 220, 317 Whipple, F. J. W., 49, 68, 112 Whittaker, E. T., 16, 17, 21, 34, 151 Wilf, H. S., 34 Wilson, J. A., 59, 99, 132, 152, 154, 165, 177, 180, 188, 191, 193, 195, 197, 198, 200, 206, 214, 215, 259, 261, 272, 274, 317 Wilson, R. L., 68 Wimp, J., 37, 84 Wintner, A., 176 Wright, E. M., 35, 241, 242 Wu, M.-Y., 34 Xu, Y., 259 Yamada, Y., 302 Yang, K.-W., 33 Yoon, G. J., 213 Zagier, D., 307 Zaslavsky, T., 35 Zeilberger, D., 34, 35, 68, 174 Zhang, R., 12, 204, 251 Zhedanov, A. S., 215, 302 Zimmermann, B., 34

Subject index

Addition formula for continuous q-ultraspherical polynomials, 249 Eq functions, 111 little q-Jacobi polynomials, 210 little q- Legendre functions, 254 Affine q-Krawtchouk polynomials, 202 Al-Salam-Chihara polynomials, 273 Almost-poised series, 111 Analytic continuation of 21 series, 117 r+" cPr series, 120 Andrews' q-Dyson conjecture, 68 Appell functions and series, 282, 283 Askey-Gasper inequality, 232 q-analogues of, 236-238 Askey-Wilson polynomials, 59, 188 multivariable extension of, 255 Askey-Wilson q-beta integral, 154, 163-168 Associated Askey-Wilson polynomials, 254 Associated q-ultraspherical polynomials, 255 Bailey-Daum summation formula, 18, 354 Bailey's four-term transformation formulas for balanced very-well-poised lOcP9 series, 55-58, 64, 365 Bailey's identity, 61 Bailey's lemma, 41 Bailey's product formulas, 219, 236 Bailey's sum of a very-well-poised 6'I/J6 series, 140, 357 Bailey's summation formula, 54, 124, 356 Bailey's three-term transformation formula for a very-well-poised 8cP7 series, 53, 364 Bailey's transformation formulas for 2'I/J2 series, 150 terminating 5cP4 and 7cP6 series, 45-47, 363 terminating lOcP9 series, 47, 263 limiting cases of, 48-53, 360-362

423

Bailey's 3'I/J3 and 6'I/J6 summation formulas, 149, 150, 357 Balanced series (and k-balanced), 5 Barnes' beta integral, 114 Barnes' contour integral, 113 Barnes' first and second lemmas, 113 q-analogues of, 119 Base modular parameter, 304 Basic contour integrals, 113 Basic hypergeometric functions, 5 Basic hypergeometric series, 1-4 Basic integrals, 23 Basic number, 4 Bateman's product formula, 221, 222 Bessel function, 4 Beta function, 22 Beta function integral, 23 Bibasic expansion formulas, 84-87 series, 80-87 summation formulas, 80-83, 328, 358 transformation formulas, 105-107 Big q-Jacobi polynomials, 181, 182 Bilateral basic hypergeometric series, 137 Bilateral bibasic series, 82 Bilateral q-integral, 23 Bilateral theta hypergeometric series, 309, 316 Bilinear generating functions, 227, 259, 281 Binomial theorem, 8 q-analogue of, 8, 354 Biorthogonal rational functions, 35, 173, 213, 257, 258, 345 Cauchy's beta integral, 132 Christoffel-Darboux formula, 200 Chu-Vandermonde formula, 3 q-analogue of, 14, 354 Clausen's formula, 103, 232 q-extensions of, 232, 235, 236, 251, 261 Connection coefficients, 33, 195-197

424

Subject index

Continuous q-Hahn polynomials, 193 in base q-l, 279 Continuous q-Hermite polynomials, 31 Continuous q-Jacobi polynomials, 191 Continuous q-ultraspherical polynomials, 31, 184 Contour integral representations of 2Fl series, 113 21 series, 115 very-well-poised series, 121-124 Convergence of basic hypergeometric series, 5 bilateral basic series, 137 hypergeometric series, 5 Cubic summation and transformation formulas, 93, 108-110

Darboux's method, 259 Discrete q-Hermite polynomials, 209 Dixon's formula, 38 q-analogue of, 44, 355 Double product theta function, 303 Dougall's summation formulas, 38, 39 q-analogues of, 43, 44, 356 Dual orthogonality, 176 Dual q-Hahn polynomials, 181 Dyson's conjecture, 68

Elliptic analogue of Bailey's transformation formula for a terminating 1O9 series, 307, 323 Elliptic analogue of Jackson's 87 summation formula, 307, 321 Elliptic balancing conditions (E-balanced), 305, 309, 313, 314 Elliptically balanced series, see Elliptic balancing conditions Elliptic beta function, 344 Elliptic Jackson sum, 347 Elliptic Dn Jackson sum, 348 Elliptic gamma function, 311, 338 Elliptic hypergeometric series, 302, 305 Elliptic integrals, 344, 345, 347 Elliptic numbers, 17 Elliptic shifted factorials, 304, 312 Erdelyi's formula, 174 Euler's identity, 62 Euler's integral representation, 24 Euler's partition identity, 240 Euler's transformation formulas, 13, 86

en

Expansion formulas (also see Addition formula, Connection coefficients, Linearization formula, Nonnegativity, Product formulas, and Transformation formulas), basic series, 19, 40, 41, 62--67, 84-87, 107,143 elliptic and theta hypergeometric series, 329-331, 337 multi basic series, 95-97 Fields-Wimp expansion, 84 q-extensions of, 84-87 q,p-extensions of, 329-331 Gamma function, 2, 21, 22 q-analogues of, 20, 29, 353 Gauss' hypergeometric series, 1, 2 Gauss' multiplication formula, 22 Gauss' summation formula, 3 Gegenbauer's addition formula, 249 q-extension of, 249 Gegenbauer polynomials, 2 Gegenbauer's product formula, 225 General basic contour integral formulas, 126 General transformations for r'l/Jr series, 138-145 Generalized hypergeometric series, 3 Generalized Stieltjes-Wigert polynomials, 214 Generating functions for orthogonal polynomials, 31, 184, 203, 259-281 partitions, 239, 240 q- Bessel functions, 31 Gosper's sums, 81, 93, 358 extensions, 81, 82, 93, 358 Hall's formula, 72 Heine's series, 3 Heine's summation formula, 14 Heine's transformation formulas, 13, 14, 359 Hermite polynomials, 4 q-analogues of, 31 Hypergeometric functions, 5 Identities involving q-shifted factorials, 6, 24, 25, 351-353 Indefinite summation formulas, 80-83, 322, 324-329, 342-344 Infinite products, 6, 20-23, 352, 353 sums of, 61, 150-152, 304-310, 312-349

Subject index Integral representations of 2Fl series, 24 21 series, 24 associated Askey-Wilson polynomials, 255 very-well-poised 81>7 series, 157, 158 very-well-poised 101>9 series, 159-161 Integral representations for q-Appell series, 284, 286, 288, 289, 294 Inversion in the base of a basic series, 25 theta hypergeometric series, 337 Jackson's product formula, 93 Jackson's summation formulas, 43, 355, 356 Jacobi polynomials, 2 Jacobi's triple product identity, 15, 357 Jain's transformation formula, 60 Jain and Verma's transformation formulas, 63, 78 Jump function, 175 Kampe de Feriet series, 284 Karlsson-Minton summation formulas, 18, 19 q-extensions of, 19, 20 Kummer's formula, 18 q-analogue of, 18, 354 Laguerre polynomials, 4 Legendre polynomials, 2 Level basic series, 111 Linearization formula for Cn(x;,Blq), 226 inverse of, 249 Linearization formula for continuous q-Jacobi polynomials, 253 associated q-ultraspherical polynomials, 255 Little q-Jacobi polynomials, 32, 181, 182, 245 Little q-Legendre function, 253 Milne's fundamental theorem, 331 Rosengren's elliptic extension of, 331, 332 Modular balancing conditions (M-balanced), 315-317 Modular group, 315 Modular hypergeometric function, 315 Modular invariant, 316 Modular parameters, 304, 312 Modular series, 316 Modular symmetry relations, 317 Moments, 175

425

Multibasic hypergeometric series, 95-97, 105-107, 112 Multibasic summation and transformation formulas for theta hypergeometric series, 325-331 Multinomial coefficients, 68 q-analogue of, 25, 68 Multivariable sums, 331-336, 345-350

Nearly poised series, 38, 39 of the first and second kinds, 38, 39 Nome modular parameter, 304 N onnegativity of 3F2 series, 232, 237 4F3 series, 239 51>4 series, 237 61>5 series, 238, 239 71>6 series, 238 connection coefficients, 197, 215 linearization coefficients, 226, 227, 257 Poisson and other kernels, 229-232, 257

Orthogonality relations for affine q-Krawtchouk polynomials, 201 Al-Salam-Carlitz polynomials, 209 Askey-Wilson polynomials, 190, 191, 206-208 big q-Jacobi polynomials, 182 continuous q-Hermite polynomials, 31, 204 continuous q-Jacobi polynomials, 191, 192 continuous q-ultraspherical polynomials, 31, 184-187 discrete q-Hermite polynomials, 209 dual q-Hahn polynomials, 181 little q-Jacobi polynomials, 182 q-Charlier polynomials, 202 q-Hahn polynomials, 180 q-Krawtchouk polynomials, 201, 203 q-Laguerre polynomials, 210 q-Meixner polynomials, 202 q-Racah polynomials, 180 sieved orthogonal polynomials, 204, 205 Orthogonal system of polynomials, 175 in several variables, 213, 255, 258

Partial sums, 58, 63, 80-83, 106, 358 Partitions, 239-242 Pentagonal numbers, 241 Pfaff-Saalschiitz summation formula, 17 q-analogue of, 17, 355

426

Subject index

Poisson kernels for continuous q-ultraspherical polynomials, 227-229 q-Racah polynomials, 229-232 Product formulas for 2Fl series, 103, 219-221, 232, 236 Askey-Wilson polynomials, 222 balanced 43 polynomials, 218-223 basic hypergeometric series, 103-105, 232-236, 251, 361 big q-Jacobi polynomials, 245 continuous q-Jacobi polynomials, 253 continuous q-ultraspherical polynomials, 223, 227, 236, 249 Gegenbauer polynomials, 225 hypergeometric series, 103,219-221,232, 236 Jacobi polynomials, 220, 222 little q-Jacobi polynomials, 210, 245 q-Hahn polynomials, 221, 246 q-Racah polynomials, 221, 246, 247 Projection formulas, 195

q-analogue of Fl (a;b,b' ;c;x,y), 283-285, 294-296 q-analogue of F 4 (a,b;c,c';X(1-y),Y(1-X)), 219, 220, 283, 290-294 q-Appell functions, 282, 283 q-Bessel functions, 30, 104 q-beta function, 22, 23 q- beta integrals of Andrews and Askey, 53 Askey, 170, 171 Askey and Roy, 129 Askey and Wilson, 154, 155 Gasper, 130 Nassrallah and Rahman, 158,295 Ramanujan, 172 Wilson, 132 q-binomial theorem, 8, 354 q-binomial coefficients, 24, 353 q-Cayley-Orr type formulas, 105, 112 q-Charlier polynomials, 202 q-Clausen formulas, 232-235, 361 q-contiguous relations for 43 polynomials, 200 Heine's series, 27 q-Cosine functions, 28 q-deformation, 7 q-derivative operators, 27, 197, 208, 251 q-difference equations, 32, 199

q-difference equations for Askey-Wilson polynomials, 199 q-difference operators, 32 forward and backward, 11 symmetric, 11 q-differential equations, 27 q-Dixon sums, 44, 58, 355 q-Dougall sum, 44, 356 q-exponential functions, 11, 12,33,34, 111, 354 q-extension, 7 q-gamma functions, 20-22, 29, 353 q-Gauss sum, 14, 354 q-generalization, 7 q-Hahn polynomials, 180, 181 q-integral representations of 2(1)1 series, 24 Askey-Wilson polynomials, 207 continuous q-ultraspherical polynomials, 185 very-well-poised 87 series, 52 q-integrals, 23, 24, 52-55, 57, 58, 65, 76, 149, 156-158, 162, 163, 166, 171, 172, 183, 185, 203, 207 q-Kampe de Feriet series, 284 q-Karlsson-Minton sums, 19, 20, 357 q-Krawtchouk polynomials, 201, 202 q-Kummer sum, 18,354 q-Lagrange inversion theorem, 107 q-Laguerre polynomials, 210 q-Lauricella function, 300, 301 q-Leibniz formula, 27 q-Mehler's formula, 275 q-Meixner polynomials, 202 q-multinomial coefficients, 25, 68 q-multinomial theorem, 25 q-number,7 q-number factorial, 7 q-number shifted factorial, 7 q-quadratic lattice, 12, 294 q-Racah polynomials, 59, 197-180 q-Saalschiitz sum, 17, 355 q-series, 4, 8, 282 q-shifted factorial, 3, 351 q-sine functions, 28 q-trigonometric functions, 28, 212 q-ultraspherical function of the second kind, 211 q-Vandermonde sum, 14,354 q-Watson sum, 61, 355 q-Whipple sum, 61, 355 q,p-binomial coefficient, 311

Subject index

q,p -shifted factorial, 304 Quadratic elliptic transformation formula, 341 Quadratic summation and transformation formulas, 69-90, 91,92, 96-100, 162, 163, 341, 361, 362 Quartic summation and transformation formulas, 94, 111, 109 Quasi-periodicity relations, 317 Quintuple product identity, 147

Ramanujan's identities, 171, 172 Ramanujan's summation formula, 52, 138, 357 Recurrence relations for Askey-Wilson polynomials, 188 big q-Jacobi polynomials, 202 continuous q-ultraspherical polynomials, 203, 204 discrete q-Hermite polynomials, 209 little q-Jacobi polynomials, 204, 205 orthogonal polynomials, 175, 200 q-Racah polynomials, 177 sieved orthogonal polynomials, 201, 202 Reduction formulas for q-Appell series, 282, 284-289, 299 Reversal of terminating basic series, 25 theta hypergeometric series, 338 Rodrigues-type formula for the Askey-Wilson polynomials, 199 Rogers' linearization formula, 226 inverse of, 249 Rogers-Ramanujan identities, 44, 241 Rogers-Szego polynomials, 210 Saalschiitzian series, 5 Saalschiitz's formula, 17 Sears' 43 transformation formula, 49, 360 Sears' nonterminating extension of the q-Saalschiitz sum, 51, 356 Sears' transformations of well-poised series, 130, 131 Sears-Carlitz transformation formulas, 64, 75, 360 Selberg's integral, 174 Shifted factorial, 2 Sieved ultraspherical polynomials of the first kind, 204 second kind, 205 Singh's quadratic transformation, 99, 361 6j-coefficients, 180, 302, 307

427

Split-poised series, 106 Squares of 2 F1 series, 232 2