Algebraic Structures and Operator Calculus: Special Functions and Computer Science

Algebraic Structures and Operator Calculus

Mathematics and Its Applications

Managing Editor: M . HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands

Also of interest: Algebraic Structures and Operator Calculus . Volume I.- Representations and Probability

Theory, by P. Feinsilver and R . Schott, 1993, x + 224 pp., ISBN 0-7923-2116-2, MAIA 241 .

Volume 292

Algebraic Structures and Operator Calculus Volume II : Special Functions and Computer Science

by Philip Feinsilver Department of Mathematics, Southern Illinois University, Carbondale, Illinois, U.S.A .

and Rene Schott CRIN, Universite; de Nancy 1, Vandoeuvre-les-Nancy, France

1

0

KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON

A C .I .P. Catalogue record for this book is available from the Library of Congress .

ISBN 0-7923-2921-X

Published by Kluwer Academic Publishers, P.O . Box 17, 3300 AA Dordrecht, The Netherlands . Kluwer Academic Publishers incorporates the publishing programmes of D . Reidel, Martinus Nijhoff, Dr W . Junk and MT? Press. Sold and distributed in the U .S .A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A . In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands .

Printed on acid free paper

All Rights Reserved ® 1994 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner . Printed in the Netherlands

To our parents In memoriam : Leo and Cecilia, Joseph and Anne

Table of Contents

Preface

ix

Introduction I . General remarks II . Some notations III . Orthogonal polynomials and continued fractions IV . Bessel functions, Lommel polynomials, theta functions V . Some analytic techniques VI . Polynomials : reference information

1 2 3 5 7 9

Chapter 1 Basic Data Structures I . Basic data structures II . Dynamic data structures

14 18

Chapter 2 Data Structures and Orthogonal Polynomials I . Dynamic data structures : Knuth's model II. Mutual exclusion III. Elements of duality theory

24 49 52

Chapter 3 Applications of Bessel Functions and Lommel Polynomials I. Analysis of the symbol table in Knuth's model II . Analysis of some concurrent algorithms

54 65

Chapter 4 Fourier Transform on Finite Groups and Related Transforms I . Basic definitions and properties 76 II . Fourier transform on finite abelian groups 89 III . Krawtchouk polynomials and finite probability theory 91 IV . Appell states 99 V . Orthogonal polynomial expansions via Fourier transform 104 Chapter 5 YoungTableaux and Combinatorial Enumeration in Parallel Processing I . Representations of the symmetric group 112 II . Clustering distributions 122 III . Inversion of circulants 129 IV . Young tableaux and combinatorial enumeration in parallel processing 136 References

139

Index

145 vii

Preface

In this volume we will present some applications of special functions in computer science . This largely consists of adaptations of articles that have appeared in the literature . Here they are presented in a format made accessible for the non-expert by providing some context . The material on group representations and Young tableaux is introductory in nature . However, the algebraic approach of Chapter 2 is original to the authors and has not appeared previously. Similarly, the material and approach based on Appell states, so formulated, is presented here for the first time . As in all volumes of this series, this one is suitable for self-study by researchers . It is as well appropriate as a text for a course or advanced seminar . The solutions are tackled with the help of various analytical techniques, such as generating functions, and probabilistic methods/insights appear regularly . An interesting feature is that, as has been the case in classical applications to physics, special functions arise - here in complexity analysis . And, as in physics, their appearance indicates an underlying Lie structure . Our primary audience is applied mathematicians and theoretical computer scientists . We are quite sure that pure mathematicians will find this volume interesting and useful as well . We expect this volume to have a utility between a reference and a monograph . We wish to make available in one volume a group of works and results scattered in the literature while providing some background to the mathematics involved which the reader will no doubt find appealing in its own right . The authors gratefully acknowledge AFCET, Universite de Nancy I, Southern Illinois University, INRIA, and NATO, for support while various parts of this work were carried out .

ix

INTRODUCTION

General remarks

In this volume, we present some applications of algebraic, analytic, and probabilistic methods to some problems in computer science. Special functions, notably orthogonal polynomials, and Bessel functions will appear. In Chapter J, we present the basic data structures we will be studying and introduce the ideas guiding the analysis. Chapter 2 presents applications of orthogonal polynomials and continued fractions to the analysis of dynamic data structures in the model introduced by J. Frangon and D.E. Knuth, called Knuth's model. The approach here is original with this exposition. The underlying algebraic/analytic structures play a prominent role. Chapter 3 presents some results involving Bessel functions and Lommel polynomials arising from a study of the behavior of the symbol table. Then, in another direction, theta functions come up as solutions of the heat equation on bounded domains, describing the limiting behavior of the evolution of stacks and the banker's algorithm. In Chapter 4 we present some basic material on representations of finite groups, including Fourier transform. Then Fourier transform is considered in more detail for abelian groups, particularly cyclic groups. Next we look at Krawtchouk polynomials, which arise in a variety of ways, e.g., in the study of random walks. The concluding Chapter 5 discusses representations of the symmetric group and connections with Young tableaux. Then, a variety of applications of Young tableaux are shown, including examples related to questions in parallel processing. Chapters 4 and 5 mainly include material from the literature that we find particularly interesting as well as some new approaches. It is hoped that the reader will find these chapters to be useful as background for further study as well as providing examples having effective illustrative and reference value. R e m a r k . The remainder of this introductory chapter provides some basic information that the reader may find useful, and is included so that the volume is fairly self-contained. Here one can find the notational conventions used in this volume; they are consistent with those of Volume 1. Many formulas and much information may be found in the Handbook by Abrajnowitz&Stegun [2], an overall very handy reference for the material of this chapter and for properties of the special functions used throughout this volume. Also, see Rainville[73].

2

INTRODUCTION

II.

Some notations

1. First, we recall the gamma and beta functions, given by the integrals r(x)= / Jo

t^-ifi-'dt (2.1)

r(x + y)

Jo

for R e x , Hey > 0. 2. In hypergeometric functions, we use the standard Pochammer notation:

(^)'^

r(a + 6) r(a)

where T denotes the gamma function. Thus, we have for binomial coeiEcients:

©=0. For example, we have (fl)n(fe)«

2-ro

n

71 = 0

= 1 + a6a; + a ( a + 1 ) 6 ( 6 + l ) i V 2 + --4. For Stirling numbers, we use Sn,k and s^^k for Stirling numbers second kinds, respectively. They may be defined by the relations:

of the first and

(x)„ = Y^S„,kx'' k=0 'Tn

x" = ^ . „ , t ( - l ) " - * ( x ) , t=o See Knuth[53], p. 65fF. (where, however, a different notation is used). 5. For finite sets E, we denote the cardinality of E by \E\ or # E .

(2-2)

INTRODUCTION

III.

3

O r t h o g o n a l p o l y n o m i a l s and continued fractions

Orthogonal polynomials arise frequently as solutions to three-term recurrence relations. A standard reference on orthogonal polynomials is Szego[81]. For our work Chihara[10] is a useful reference as well. 3.1

ORTHOGONAL POLYNOMIALS

Given a bounded, nondecreasing function F{x) on R , we define a corresponding Stieltjes measure dF{x). The moments of dF are defined by

We will take F to be normalized so that ^o = 1- It is assumed that the moments exist for all n > 0. Integration with respect to dF, m,athem,atical expectation or expected value , is denoted by angle brackets ( • ) {f(X))= r

f(x)dF(x)

(3.1.1)

J — CO

in particular, we can write the moments as Hn = {X"). In the discrete case, we have a probability sequence (discrete distribution), {pk }k>o, satisfying p* > 0, ^ p ^ t = 1, with corresponding moments /^n =

Y^(ak)"pk k=0

where at denotes the value taken on with corresponding probability pk- In the (absolutely) continuous case, we have a nonnegative density function p{x), and fJ'n =

I x" J — oo

p{x)dx

We define an inner product (•, •} on the set of polynomials, which is the mathematical expectation of their product. On the basis { x" }, this is given in terms of the moments: {X",X"'} = ( X " . X ' " ) = /

X^-X"" dF{x) = fXn+m

J — oo

The corresponding sequence of orthogonal polynomials, { <j>„{x) }n>Oi say, may be determined by applying the Gram-Schmidt orthogonalization procedure to the basis { a;" }. The polynomials { n{x) } satisfy M^)^m{x)dF{x) = n"'

^ , ^

(3.1.2)

4

INTRODUCTION

where 7„ are the squared norms. We normalize 70 to 1. With (f>„{x) = x" + • • •, monic polynomials, the { <j>„{x) } satisfy a three-term recurrence relation of the form: XTi= 0. Choose e so that R~ e > 1. • Because of this, one looks for the singularities of functions when it is known that the coefficients a„ grow as n —» oo. For our purposes, the following suffices. 5.1.2 L e m m a . Let f{z) = S ^ o * * " ^ " ^'^ meromorphic, analytic in a neighborhood of 0 in the complex plane. Let ^ denote the pole of f nearest to the origin. If ( is a poJe of order p, then we have the estimate ,

,

A

nP-i

r(p) c"+p where A = lim f(z){z

Proof:

— C)*".

Write g{z) = f{z) • {z - CY- So A = g{0 and

where the second term has at most a pole of order < p. Now, for \z\ < |C|, the binomial theorem implies

At a pole of larger modulus, the same estimate gives a rate of growth exponentially slower, • whereas terms of lower order poles grow at correspondingly lower powers of n. See Flajolet-Vitter[28] for more along these lines.

8 5.2

INTRODUCTION CENTRAL LIMIT APPROXIMATION

The central limit approximation is a detailed application of the theorem of DeMoivreLaplace that one can approximate sums involving binomial coefficients, interpreted as taking expectations with respect to binomial distributions, by Gaussian integrals. Here is a useful version. 5.2.1 L e m m a .

Then, as N —* oo,

Let f be a polynomial.

Another formulation of the same statement is:

N \ /' N k + N/2) ^ \N/2 where k = avN.

In particular, we have

5.2.2 L e m m a .

For p > 0, as iV —> 00,

2-2"'

y/Nda

2 / ^ 1 / 1

.,4-',,^ |*|n,4>m) = j

'i>nix)(j>mix)dF{x)

=Jn6„m

J —00

where 6nm is Kronecker's delta equal to 1 if n = m, 0 otherwise, and 7„ are the squared norms. 2. The moment generating function denotes a generating function for the sequence of moments {^n }• Note that for symmetric distributions, where the odd moments vanish, we often write the generating function for { n^n }• 6.1

POLYNOMIALS CORRESPONDING TO DISCRETE DISTRIBUTIONS

Here we list basic information on polynomials orthogonal with respect to the binomial and Poisson distributions, respectively. • Binomial distribution — Krawtchouk polynomials { Kn{x) } Recurrence formula X Kn - A'„+i + n{N - n + l)A'„_i with A'o = 1, K\ = X.

10

INTRODUCTION

Measure of orthogonality and squared norms

{Kn, Km) = 2-^^ ^

jn =

( , )K,,{2k

- N)Kmi2k

- N) = y„ 6„^

n\{-in-N)„

Moment generating function V —r/^n = (cosh 5 ) ^ Generating function for the polynomials N

T ^ K„(x) = (1 + t,)(^+^)/^(l - vY^-^y' n=0

Formula for the Krawtchouk polynomials

lt=0

• Poisson distribution — Poisson-Charlier polynomials { P„{x,t) } Recurrence formula xPn=

Pn+l

+ (< + n)Pn

+

tuPn-l

with Po = 1, Pi = a ; - t . Measure of orthogonality and squared norms {Pn, P,n)

=e-'

^ 4=0

-

Pnik)Pm{k)

= 7 „ Sr,

•

7n = n\r Moments /in =

Y^Sn^kt'' k

Moment generating function V n=0

—Ain = e x p ( i ( e ' - 1 ) ) n

INTRODUCTION

11

Generating function for the polynomials

n=0

Formula for the Poisson-Charlier polynomials

P„(x,0 = (-ir^(")k-^\ + (Ik^k + dk-\ = 0, <j>o = 1. Thus, the {A } satisfies X"0 = /^Cnkk fc=0

26

CHAPTER 2 Proof:

Inductively, assume the relation, which holds for n = 0. Then X""'"Vo = ^c„i(it(/>jfc+i + qkk + dk4>k-\) k = 2 _ , ( * * - l '^nk-l k =

by Prop. 1.1.1.

+ qkCnk + dk+\

Cnk+l)^k

}_CnJr\kk k

•

This shows X " as the n-step transition operator. Shifting all indices by k 1.1.3 Corollary. Denoting in n steps, we have

hy Cni,k the number of ways of going from level k to level I n

X''^k

= Y,Cnl,k(f>l 1=0

with coi,k = 1, ifi = k,0

otherwise.

We can introduce the monic polynomials J/J^ = ;t ioh • • • ik-irelation xil^k = i>k+i + Qki^k +

They satisfy the recurrence

ik-idki>k-i

Equation (3.2.1.2) of the Introduction indicates the link with the continued fraction approach of section §1.6. As follows by equations (3.1.3), (3.1.4) of the Intro., the squared norm of ^k is Ik — toil • • • ik-i

didi

•• -dk

As in §3.1 of the Intro., eq. (3.1.2), for the inner product we have 1.1.4 P r o p o s i t i o n .

The {(fek)

satisfy = k

«'•«-{?• j ^k with the squared

norms 2

ek = Ukr =

dxu2 * • • dk iQii

••

-ik-i

Thus, the markovian model indeed corresponds to the notion of Markov chain or random walk in this case. Let us rewrite the above corollary in terms of histories. We have the result (Proposition 7 of Flajolet, Fran^on &Vuillemin [26])

DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS

1.1.5 T h e o r e m .

In

The number of histories Hk,i,n is given by the expected

27

value

particular, Ho,k,n = {X"^k}/£k,

Hk,o,n = {X"<j>k}

with Ho,0,n = {X"') = fin the

moments.

And we have the observation, from the point of view of operator calculus, via the first statement of the Theorem, 1.1.6 Corollary.

The exponential

V

generating

^Hk,i,n

functions

= (e'^ k(t>i)/ei

are the matrix elements of the operator e'^

in the basis {(i>k}-

R e m a r k . This feature is a connection with Volume 1 [21] of this series, Chapter 7 in particular, that will be useful in our analysis. It turns out that, appropriately modified, these techniques can be applied to the study of Knuth's model as well. 1.1.1 Enumeration

of histories in Knuth's

model

Here we introduce c^j., the number of paths starting from 0 that are of height k after n steps, with s the number of insertions and negative queries combined. Prop. 1.1.1 becomes 1.1.1.1 P r o p o s i t i o n .

The c^f. satisfy the

recurrence

< + i k = ^ < " * - ! + s c^-1 + qk c;^ + dk+i c'„ ;t+i with CQO = 1, CQJ. = 0, A:,3 > 0. The idea here is to introduce the generating function

Cnk{t) = Xll"*^"* 8= 0

Thus, the s is summed out, leading to a situation similar to the maxkovian model. We see immediately that

28

CHAPTER 2

1.1.1.2 P r o p o s i t i o n .

The C„kit) satisfy the

recurrence

Cn+l k = tCn k-1 +{t + Qk) C„k + dk+\ Cn k+\ with Coo(t) = 1, Co kit) = 0, ^ > 0. The t indicates t in the case when there axe negative queries, otherwise it is omitted. Here we introduce the operator Xt acting on a space with basis { (j>k{x,t) } such that n

k=0

as in Prop. 1.1.2. Dual to Prop. 1.1.1.2 is 1.1.1.3 P r o p o s i t i o n . The polynomials { 4>kix,t) } satisfy the

recurrence

X(j>k = t(f>k+i + (t + qk)4>k + dk4>k-l with 1. Prop. 1.1.1.3 gives, where we rec?i,ll that the middle term, involving (pk, corresponds to queries: 1.2.1.1 P r o p o s i t i o n . 1. The recurrence

For priority queues we have:

relations X<j>k = i(f>k+l + k-\

with the squared norms ek{t) = k{x,t)=t-''I^Uk{xl2yft)

Tchebychev polynomials

of the second kind, and the moments

/^2n(0 =

1. Thus, 1.2.2.1 P r o p o s i t i o n .

For linear hsts we have:

1. The recurrence reiations x4>k =t 4>k+i + k4'k-i with the squared norms £fc(e the number of histories of length n = p + q + r with p insertions, q negative queries, r = r\ -{- r^ where ri (resp.r2) is the number of deletions (resp. positive queries). Consider the following generating function :

^(M,.) = E Ek-, [t,t + At]) = XkAt + o(At) P(Ek+i

-^ Ek] [t,t + At]) = fikAt + oiAt)

P{Ek -^ D; [t, t + At]) = akAt + o(At) We have then: Pk{t + At) - Pk{t) = {-{ak

+ \k + tJ.k)Pk{t) + \kPk-i{t)

+ / i t P t + i ( i ) ) At + o(At)

Dividing by At and letting At —> 0, we have the system of differential equations; Pl{t) = ~iak + \k + ^^k)Pk{t) + XkPk-iit)

+

fikPk+iit)

with boundary conditions : Pi'(t) = -{ai

+ fi.i)Piit) +

Pl,{t) = -{an

+ )ln)Pn{t)

fiiPiit)

+ A„Pn_i(t)

Writing P ( t ) as the row vector with components Pk{t), we can write the system in the form P'(t) = P ( t ) M where the matrix M has columns ( . . . , A i t , - ( Q t + \k +

fik),fik,---y

Note that the ajt are the infinitesimal rates at which the system deadlocks. See Feller[22] for thorough background on Markov processes and how to analyze such systems.

52

CHAPTER2

2.2 APPROACH VIA RECURRENCES Here we can apply the transition approach of Section 1 as well. Assuming that access/release of a resource is independent of the time, n, and that access/release depends only on the number of active processors, we have a recurrence of the form Hk,n+1 = dk+-iHk+l,n

+ qkHk,n

+

ik-lHk-l,n

where Hk,n is the number of histories such that k processors are active at time n. A deadlock is modelled by the fact that a path touches the a;-axis, i.e., there are no processors active. To calculate the behaviors, we could enumerate all histories whose height is strictly greater than 0. This is like the problems studied in Section 1 and similar methods are applicable. Karlin-MacGregor[48][49] have studied the relationship between birth-and-death processes and orthogonal polynomials. For studies from the point of view of concurrency measures, see Arques, Frangon, Guichet&Guichet[6] and Geniet, Schott&Thimonier[37]. III.

E l e m e n t s of duality t h e o r y

Here we briefly indicate the general approach to duality — the correspondence between the recurrence for the c„jt and the recurrence for orthogonal polynomial basis vectors used in §1.1. The idea is to interpret Cnk as components of a vector C „ . You have a discrete dynamical system of the form C „ + i = ACn with C „ a vector of components Cn(k). operator X is determined by the relation

This gives a matrix C{n,k)

k

where $ is a column of the basis vectors j. Thus

And X " + V o = X{X"4>o) = C „ • X $ = ACn • * = C „ • A*$ So the action of X on $ is dual to that of J4 on C: X^

= A*^

— C„(/;).

The

DATA STRUCTURES AND ORTHOGONAL POLYNOMIALS

53

Example. Consider the factorial powers x^"' = x{x — l)(x — 2) • • • (x — n + 1). Then the Stirhng numbers of the second kind are determined by the relations C" = y^^Sn^kX k

With 4)k = x'*', we have the action of X X(f>k = ix-~k

+ k} xk

By duality, this gives the recurrence for the Stirling numbers Sn+l,k

= Sn.k-l

+ ^•Sn.lt

R e m a r k . Basic studies concerning Knuth's model are Frangon, Randriajiarimanana&Schott[33] and Randrianaximanaj:ia[74]. The probabilistic approach is taken in LouchEird[58], and Louchard, Randrianarimanana&Schott [60]. See Maier[63] as well. A basic reference for this chapter is Flajolet, Pran5on&Vuillemin[26]. Also see Flajolet[24], and Fran5on[30]. For various applications in combinatorics via an approach similar in spirit to that taken here see Godsil[38]. Meixner discusses the polynomials now known as those of Meixner type in Meixner[69]. A presentation of finite operator calculus is given in Rota[78].

Chapter 3 A P P L I C A T I O N S O F BESSEL F U N C T I O N S A N D LOMMEL POLYNOMIALS

In this chapter we present some applications of Bessel functions. Section 1 is devoted to the symbol table in Knuth's model while, as a complement, section 2 discusses the rate of convergence of zeros of Lommel polynomials to those of the corresponding Bessel function. Section 3 concerns the analysis of some concurrent algorithms, particularly some aspects involving special functions. I.

A n a l y s i s of t h e s y m b o l table in K n u t h ' s m o d e l

The approach developed by Flajolet, et al., leads in this case to a divergent generating function. So a different approach was called for. Recall that for Knuth's model, one considers the generating function Hn(t) = Yli'^'/^^•)-^n where s counts the number of insertions. A study of the numbers H „ ( l ) as a first step towards the solution was made in Flajolet-Schott[27] where the asymptotic behaviour of these numbers was obtained with the help of Bessel functions. We recall these results in the next section. 1.1

BESSEL NUMBERS

Here the notation H„ refers to iif„(l), i.e., the function H„(t) evaluated at t = 1. Flajolet&Schott[27] proved that:

Hn^Y.

n+2

as n —> oo

lt>l

/f„Ri(n/2elogn)" R e m a r k . The numbers H„ are denoted 5 * in [27]. The starting point for the study is the fundamental recurrence for Bessel functions : J^+i{x) = 2vx~^ Jv{x) —

Jy-i{x)

Rewriting this relation as: ^-J^^

= 2vx-'

-ll{Ux)IJ^+,{x))

and iterating the process, setting x = 2, yields

This continued fraction provides the connection with symbol table histories for the analysis of this data structure. We now recall some of the results from Flajolet-Schott[27].

BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS

1.1.1 P r o p o s i t i o n . The asymptotic Bessel functions is expressed by

expansion, as u —* oo, of a quotient of consecutive

vj,{i)

1.1.2 L e m m a .

55

Z ^ " ' " ' " pn+2 n>0

We have J.-i(2) vJ^{2)

1 1V

"'•

r>0 ^

'''•

where

and the Bessel numbers Hn

satisfy

The asymptotic behavior of the H„ axe deduced from these results. 1.2

SYMBOL TABLE AND BESSEL FUNCTIONS

For the symbol table in Knuth's model we have the recurrence, cf. Ch. 2, Prop. 1.1.1.3, X(f>k = k + (f>k-\

with squared norms ek{t) = ||^jt|P = ^""*1.2.1 P r o p o s i t i o n .

The solution to the

recurrence

X(j>k = t(j>k+l + k^k + k-l with -\ = 0, 1^0 = 1 may be expressed in terms of Lommel

polynomials:

4>k = r * / 2 i ^ , , _ , ( - 2 ^ / < )

Proof:

Observe that (l>k = t'~''''^ipk{x/\/i)

where the polynomials V"* satisfy

xipk = ^k+i + kt~^l'^il)k + V"*-! Comparing with Intro., eq. (4.1.5), yields, with e — -y/t, ^k = i i t , _ , ^ ( - 2 v ^ ) from which the result follows.

•

R e m a r k . In the following, we will denote the argument —1\fi of the Bessel function by r . An important feature comes via Intro., eq. (4.1.6).

56

CHAPTER 3

1.2.2 Definition. is denoted by E^.

For a given r , the set of zeros of the function of x given by J _ i _ i ( r )

Now we have 1.2.3 L e m m a .

Let (^ € E^ be a zero of J-I^X{T)-

Rn.-dr)

Then

J-C(r)

Proof: Replacing in Intro., eq. (4.1.6), ex by x and the argument —26 by r , we have, dividing out J-^iT), ^"•-^^^) -

and hence the result.

- j l A ^

+

J_.(r)

^"-1-1-HO

•

It turns out that t h e set S r will correspond to the values of x where the measure of orthogonality for the (j>i: is concentrated. From the recurrence relation we see that one thinks of the values of x as eigenvalues of the matrix operator given according to the right-hand side, hence we have the spectrum S^. To see this, we use the continued fraction, cf. Intro., §3.2, which gives the moment generating function, M{3) = ((1 — sX)~^), as we saw in Chapter 2. Here we have M{s) = i l l - t s ^ l - s - ts'^/l

-2s-

ts'^/l

-3s-

ts'^f • • •

corresponding to the monic polynomials associated to the (^t, satisfying the recurrence xtpk = ^k+i + krpk + i^k-i, as noted in the remarks preceding Prop. 1.1.4 in Chapter 2. In particular, both sets of polynomials have the same measure of orthogonality. The Bessel numbers of the previous section show that as a power series at zero, it has radius of convergence equal to 0. However, the continued fraction gives the appropriate analytic continuation to regions corresponding to existence of the Stieltjes transform, which is related by the transformation s~^M{s~^). 1.2.4 T h e o r e m .

The moment generating function has the formal

Mis) = ( - ^ ) = I-SX with T = —2y/i. The Stieltjes transform is

^

differentiation

^

sVtJ-l-l/s{T)

^s-X' ViJ-i-,{T) giving the spectrum S^ as the zeros of the denominator. measure is given by the residues

the prime indicating

^

expression

with respect to s.

The corresponding

probability-

BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS

57

Proof: From Watson [84], pp. 153, 303, as noted in the previous section, the ratio of Bessel functions can be expressed as a continued fraction which yields the equation

^ - ^ = 1/—+ 1 Z

Ji,\Z)

'

1/—+2

Z

Z

I

1/—+3

Z

Z

I

Z

1 Z

And thus, •

2 / 1

„

±2

f-\

J^^

o„

/i

A,*!

o_

1 - — = ts^ / I - 3 - ts^ 11 - 23 - ts' 11 - :ia - ts Dividing numerators and denominators by s y i we have

Setting z — —2\/t, v = —1/s, we have

M~

2v \ z

J^{z)

We get the Stieltjes transform, {{s — X)^^),

) ~

2v

via s~^M{ljs).

J^{z) •

R e m a r k . Observe that the Bessel function in the denominator satisfies

(-y^)-._,_,(.) = X : ; ; T r ^ SO the spectrum is given by the zeros of the function

E

n=0

^

'

Now we see that the discussion in [27] is for the case t = \. For general t we have the asymptotic form of the zeros / •

^"^"

< "

n\{n + \)\

as n —• oo. 1.3

INTEGRATED COST F O R T H E SYMBOL TABLE

We follow the technique used in Chapter 2 for priority queues. We want an expansion of (1 — sx)^^ in terms of the polynomials (/>„. Write ( ^ - I ^ ^ n W ) = M{s)V„is)

(1.3.1)

58

CHAPTERS

We know that (t>o = 1, 0 by orthogonality. Introduce the operator A j acting on functions f(s) by A,/(s) = The principal feature of this operator is that A , ( l - sx)~^ = x{l -

sxy^

that is, the functions (1 — sx)~^ are eigenfunctions of A, with spectrum x. Applying As to both sides of eq. (1.3,1), it follows via the recurrence for 4>n that, for n > 0,

Applying eq. (1.3.1) to the left side of this equation yields M{s)

{tVn+l{s)

+ nV„{s)

+

Vn-x{s))

Thus, Vn{s) satisfies the same recursion as does i^„ except for initial conditions. Setting n = 0 gives, using 4>i — x/t, ,

1

y,

M{S)-1

via the operator As, while from the right-hand side we have

Therefore

From this one guesses the general solution

K(.) = * - " / ^ ^ ^ ^ i ^ cf. Intro., (4.1.6), which is readily verified. We thus have 1.3.1 L e m m a .

The expansion in Lommel __L_ -

•'•

\^.^n/2j.

polynomials /^\

Jn-l/sJT)

BESSEL FUNCTIONS AND LOMMEL POLYNOMIALS Proof:

59

From the above, we have the expansion 1

°°

J — ^

= Mis)

J2

Vn{s)M^)/£n

with the squared norins Cn = * - " . Now,

s\/iJ-i-i/^{T)

and hence the result.

J-I/,(T)

•

Now consider positive queries at cost fc on a file of size k. We have ^oCoiRV)4>n

= n^4>n

Set s^/i

J_i_i/s(r)

so that (1 — sx)~^ = J2 ^n(s)4>n(3;). Using the second main formula for integrated cost, Ch.2, Theorem 1.4.5, we want to calculate oo

3{{i - sXYHoCo{RV){\ - sxy^) = s J2 Wmisfmh-"^ using the orthogonality relations of the „• Substituting back in the expression in terms of Bessel functions for Wm(s) yields oo

'>m-\ja\J)

-T St

J-\-\IS{T) ^

with Er the poles of the summands. So we look at an expansion in I/5 of

E

Jm-sir) \.J~x-.{r)

expanding in partial fractions the main contributions come from the terms with double poles: / Jm-s{T)

\

_

• ^

Jm-

y(s-cy

+ lower order

The Stieltjes transform says that

(7:^)=^ ''~^'

p(C)

C^.'-^

_

J-S{T)

ViJ-i-.(r)

60

CHAPTERS

with

Hence 1 Y^

Jm-