Combinatorial identities;: A standardized set of tables listing 500 binomial coefficient summations

、、

，，EE--·· 飞、

nk ••• ,,E

r

Combinatorial Identities

H. W. Gould

Copyright @ Henry W. Gould 1972 Printed by Morg:mtown Printing and Binding Co.

To my wife Josephine for her never-failing support and enthusiasm "How do 1 love thee? Let me count the ways."

Combinatorial Identities A STANDARDIZED SET OF TABLES

LISTING

5∞ BINOMIAL

"Sc仿ηtia η.on

COEFFICIENT SUMMATIONS

habet inimicum nisi ig n.o rantiam"

HENRY

Pr时essor

W.

Go皿D

of Mathematics

West Virginia University

Revised Edition Morgantown , W. Va.

1972

INTRODUCTION

Anyone who has taken the time to read far into the vast

1iterature of mathematics wi11 be aware of the fact that summation formulas for binomial coefficients are very wide1y scattered in books and

journa1s.

咀le

situation is para11e1

wi世1

what we

shou1d have if no tab1e of integra1s existed for our dai1y use and convenience. gap 由ld

It is our object here to atternpt to fi11 this st由ldard

we present what we be1ieve is the beginning of a

set of tab1es for binomia1 coefficient summations. 、

..

It wou1d , of course , be a hope1ess task to 1ist a11 known 。r

a11 possib1e formulas , and so the best which an aspiring tab1e-

maker may hope to achieve is to 1ist at a given date those formulas which seem of frequent occurrence , or which are

interes1址ng

some specia1 reason , or which are very genera1 in nature.

for It might

be possib1e to produce some type of minima1 1ist of formulas together with a set of operating ru1es whereby other formulas could be obtained , yet we sha11 not attempt this as such an

att四Ipt

wou1d destroy the uti1ity of the kind of tab1es we propose. Certain 1ists of identities which exist in the 1iterature deserve

specia1 mention

here.

Dougal1 , and Bateman seem very

世le

1ists of Hagen , Netto , Schwatt ,

iu飞portant.

Project , which resu1ted in the series on

咀le

Bateman Manuscript

specia1 到mctions ，

is a

monumenta1 achievement in bringing about order in a chaotic fie1d. Bateman himse1f had projected a book on binomia1 coefficients apart'

from their connection with the hypergeometric functions. The author has worked his "shoe box" notes into pub1ishab1e form.

Wh en

a tru1y comp1ete index of formu1as is conternp1ated , such notes and materia1 w土工 1 be taken into account.. See bib1iography be1ow. - i

-

su回nat10nø

Binom1a1 coeff1c1ent

heve a1weys proved a popu1af

Bubject 1n prob1em departments of mathemat1ca1 journals.

Much

lnterest was stirred up a1so after A. C. D1xon f1rst found a c108ed expression for the sum of an a1ternatlng eeries of cube8 of b1nom1e1 coeff1cients. Thi8 1ed to the great contribut10n8 of ::'

", .omial

~'ranel ，

and Mac lt.ahon on the eummetion of powers of the

coeff1cients. Meny newcomers to this work are

,

however

,

l' Inhrm(diaire 些主 Mathémat1ciens.

unawere of Frane1's paper 1n

The author' s 1nterest in th 1s work began- twenty-seven years with the study of prob1ems 1n the

ag。

!m盯 1can Mathemat1ca1 坠且且1..

!t would be 1mposs1b1e to g1ve complete references for the tab1es of formu1as he has

,

collected and 1n some caees extended for he would ,

have to 1iet v1rtual1y a11 the journa1s which have been at his disposa1 , and this wou1d inc1ude journa1s and books at theUniversity of V1rg1nia

, Duke

Univere1ty , Univers1ty of North Oar011na , the Library

of Congress , and West Virgin1a University.

The author was warned

10ng ago that he might we1l f1 0u nder in formu1as , 1f on1y 1n one 8ma1l field , such as St1r11ng numbers , or Bernou111 numbere , not to ment10n this vast subject of b1Dom1al coefficient summations , and that is one reason he w111 feel some 8enee of accomplishmen"t if even a beginning 1s made in the present tab1es. It must be poinhd out that we omit proofs 1n the present work. In most 1nstances the author ha8 worked out proofs wh1ch

,

dl何时 from

the known proofs; however to 1nc1ude proofs in the present

tex乞 wou1d

magn1fy the 1ength of thie volume by about ten. The 8uthor's per80nal interest 1s not the mere tabu1ation of for皿11a8 ， but th18 emphaeis 1_ deslred in the tab1es we g1ve here.

-

11幽

PROOFS As far as proofs are concerned , it is important to indicate the 。f

most 缸nportant

the fo110wing:

methods. c。η飞parison

Th ese seem to be any one or a cornbination

of coefficients in series expansions ,

use of differentiation or integration , finite àifference rnethods , solution of difference equations or recurrence

f，。ηnu1as ，

mathernatica1 ,1

induction , enurneration of 1attice points , theory of combinations an卢 permutations ，

and generalserics transformations.

the finite Taylor's series for po1ynomials is

Th e use of

rather 却nport田lt

a1so.

1n the last part of our ihdex wi11 be founct some useful transformations which are especial1y important in handling the algebra of series. Many of these wi11 be found 1n intricate detai1 1n Schwatt's Introduction to the Operations wi th Se rie~

[7 2 J

NOTAT10NS AND DEFINIT10NS 1n the body of our tables , we have used the 1etters x , y , z for real nurnbcrs except 1n a few instances appears.

哑le

letters j , m, n , r , s

negat1ve intcgers.

1n most cases

here some note to the contrary

",

generally have been reserved for nonk

has been used for the dummy variab1e

of summation and appears in no other usage.

Th e bitiorr归1 coeff1cient

(X J 飞 n.l，

is defined by the fo110wing:

(:)=川);;… 4打 (1 =

x! r (x + 1) in a ;f ew cases , but general工y we and we define feel it best to avoid use of the standard gamma function. The gamma function of Euler i6 only one of an infinity of ways of extending the factorial concept. -ii1-

l

』/

x-fIt--

=、

/flt 飞

1lt/

飞

lll-/

、11ftffιln

Z/fttmv -x-

EE nxn

、飞

\fh/ xn x-mH

xn·

、、 //tII

/i\μ1\

By extensive use of re1etions such e8

(-OE(-1)nf:-) 叫ζ

nhEl--/ ny n n』 J

aEE

E

n

，，，『 tz飞

\}/ 、‘.，，

Aεn

fftt\

n

many veriations of the listed formules We give msy be found. A few of the more

usef、u1

binoruia1 coefficient relations of this type ere col1ected >>

together 1n our lsst tab1e. Although we do not include formu1as which sum seriea involving Bernoul1i numbers , Stir11ng numbers , Eu1er numbers , etc"

We do need to

define the Stirling numbers ineswuch ss many sums involving binomiel coefficients can on1y be given in terms of these. We sha11 define thE Stirlinz 凹旦坦王旦旦主旦旦旦旦主坠旦立 by theeoeff1eient

cZin the

n

expansion

C)- ~_ C~

x

k

The Stir 1i n 旦出业旦旦旦生旦旦旦旦主丛丛 we define by the expension n

xn

-

.2;

11111 ,,,XK

nk ZEE-

nu

,

J

k-=O See the

auth町 's papers(sec。nd bipli。 graphy)U 汀，巳 8] ， [3吁， (41) ，巳吨，

(621 for discussions of notation and other properties. A better nota tion S 1 (n ，时， S2(n , k) was introduced in[11]. -l. V-

工n

particular it followa that

哇=叫 :ω(

j嚣'

k \

J=olj / and

时 CB=(-1)k

(*)

AIK+a1IK-n 、 y l l l lJ-dd L 1,. _ .: I l ,. , .: I j! - j

j::O 'k - j I

飞 k + j

I

'

the latter being perhaps the simpleat expression known for the Stirling numbers of the first kind , given by L. SchlBmilch.

Sch工If f工 i

and O.

The reader should consult [2汀， [3才， [3吁，也才， [61J

for details regarding the Stirling numbers. 工n[3司 the author proved a formula inverse to (*) above. 工 terated

summations of terms have been omitted , since their

inclusion would necessitate a tabulation of much greater length. The receþtion accorded the first printing (1959-60) of these tables has been so overwhelminglyfavorable and continued ,‘ thælt the demand for copies has long since exhausted the supply. This corrected reprint has been undertaken to meet the need for such tables. Rather than retype the entire manuscript and run the risk of introducing new errors , it has been decided to correct those misprints which are easy to change. At the end will be found a set 。f

errata tables , bringing'together those errata which have been

ca 工 led

to the

attent 工 on

of the author in the intervening years.

These lists will be useful to anyone who has access to one of the 。 riginal

printings of these

altered; however the reader can

a

工ist

consu 工 t

of research since 1960.

tab 工 es.

The bibliography has not been

of the author's pàpers is included so that those items for a general idea of the trend Most ofthis work has been supported by

research grants from the Nationa 工 Science Foundation since 1960. A few new formulas have been introduced , just those which seem to be absolutely essential.

Al工 binomial

formulas are kept on

file in a card catalogue , and proofs are kept in running volumes of notes now covering several thousand clear that this reprint will serve a author completes a

manuscr 土pt

treatise on the subject of

pageß_

工t

is abundantly

usefu工 purpose

in preparation for a

combinatorial

-v-

until the comp 工 ete 工 y

土 dentities.

new

Sucha new treatise will offer many new identities , a complete revision of the present tables , model proofs , hundreds of references to the literature and a whole chapter on the generalized Vandermonde and Abel convolutions which have occupied much of the author's attention. obl 土 ged

The author is

to Drs. T. A. Botts and Reed Dawson

who originally suggested the compilation of the tables. He must express his deepest thanks to the following who have made many suggestions

and

or suggested new

土n

many cases supplied corrections to the tables L. Carlitz , John Riordan , Donald Kn uth ,

formulas~

Eldon R. Hansen , R. E. Greenwood , Michael P. Drazin , Verner E. H'oggatt , Jr. ‘

,

Julius Lieblein , Emory P. Starke ,

and many others

too numerous to mention. The author would be pleased to hear from users of the tables , and welcomes any

summat 土 on prob 工 em

involving binomial coefficients

whether the user knows the sum or not. improvement will be

careful 工y

Al 工

suggest 土。 ns

for

studied.

The attent 土。 n of the reader is called to a very important recent book COMB工 NATOR工 AL 工 DENTITIES ， by John Riordan , published by John Wi 工 ey and Sons , New York , 1968. This is the first booklength treatise 土 dentities.

deal土 ng entire 工 y w土 th

and operators. Anyone

,

generating functions ,

dea 工 ing

Riordan's book. Some of this re 工 ations

wr 土 ter's

of books , in

the

work on inverse binomial

is treated in detail in Riordan's book. There has

theory , but the subject of i口

partitionp 。工ynomials ，

with our subject matter should consult

been a great flood of books recently scattered

combinatoria 工

The subject matter covers: recurrences , inverse

relations (two chapters)

series

the theory of

1土 terature

unpub 工 ished

dea 工 ing

with combinatorial

comb 土natorial 立 dentit 主 es

remains

of several hundred journals , in portions

manuscripts and files , etc.

expected that this situation will shortly change. for pedagogical treatises on the

subject.

工n

There

工t

土s

is a need

TIl E ART OF COMPUTER

PROGRAMMING , by Donald Kn uth , Addison-Wesley , 1968 , V。工 .1 ， pp.53-73 , the reader will find a attacking a

prob 工 em

good ，工 eisurely

account of techniques for

of summing a binomial series. -vi-

ABαJT

A NOTE

THE SYSTEM OF INDEXING

It is importent 1n 8 compi1etion such 88 this tomake use of 8 system of 1ndexing to

8110~

~hich ~i11be

fe~

the eddition of e

e8sy to use 8nd flexib1e enough

formulss from time to time without

complete1y destroying the old ordering. We therefore drew Up e sequence of tebles of formu 1es , numbered 。， 1 ， 2 ，}， etc. Eech teb1e lists formulSs of the type S:p/q for

certe1n ve1ues of p end q , where p denotes the number of binomie1 coeff1c1ents eppeering in the genere1 numeretor term of the summetlon

,

8nd q hes the S8me me8n1ng for the generel denom1netor term. The tebles ere to be found errenged 1n the ordering 。 /0 ， 1/0 ， 0/1 ， 2/0 ， 1/1 ， 0/2 ， }/O , 2/1

,

1/2 , O/~ ， 4/0 ，当V1 ，

.• .etc.

It 18 eesy to prove thet the teb1e number , n , essigned to eech type formu18 , S:p/q , 1s given uniquely eccording to the formu1a p+ q D-q+L

k-q+ 灿 q)(p +什 1 )

k-O Thus , in this system , e11 formules for sums of form s:4/} ehou1d be 11sted in teb1e number ~1. The proof ~f th1s is releted to the usuel proof of the countebi1ity of the r8t1one1 numbers. Conversely , if We desire to find out whet type Qf

s\皿 18

tobe

found in e teble of number n we mey determine' th1s from the formu1as

p- 告(N-1)(N+2) - n where

q&n 回去N(N-1)

N_G1+ 巧有J 丁J

' end where

,

[寸 meens the

1ntegr e 1 pert of x. The proof depends on the fect thet every odd squere is of the form 80 + 1. Thus for teble ~1 ， N

-= }5 - ~1 - 4 , eod q

= ~1

-部的 (7)

= }1

-v 工工-

=8 ,

so

- 28 - }.

p-= 告(7)( 10) - ~1

In this

sec∞d

eòition of our formulary , the followlng

tebles Bre 11sted Bnd the indiceted eporoximete number of summBtions of specified type Bre to be found within eech +.8 ble • • :",_

7.E NR.

Form S:p/q listed

t 2

2/0 1/1 。/2

5 6 7 8 10 11 12 16

A1//2O

17

当/2

21 22

6/0 5/1 4/2

;/0 2/1

48 9 3 8 3 5

;/1 2/2 4/1

225 1+

2

75A6/ /冒

;1 71 97

z

135 26 180 29 2 52

1/0 。/1

岛主

x

Approx. nr. sums listed

2

F 56

15 30

p/O Generel theorems TOTALI

555

As might be expected , the 6ums of form 5:2/0 , 5:1/0 , 5.2/1 , end

S:;lo

are most well known. 1n general 6ums of the form 5.0/哇'

with lerge veluee of q ,

ere relatively rare in the literature.

An important fector to keep in mind 18 that there 18 no unique form Srp/q in which e formula of Some bapic type 1s elwsy8 wr1tten , bec8use by use of elementsry binomiel coefficient identitie8 we may rewrite a given formula uelng more or fewer binomial coefficienta. Thus there will slweys exist 8 large degree of whet i8 really cross-classification. However we do not propose to draw up a minimal list of formulss. An axiomatic study is planned later. -viii-

TABLE 1 SUMMATIONS OF THE FORM

Sr1/0

00 c--飞

)‘

(1.1)

f l IX 飞

‘,

zk a(1+z)X

Binomiel Theorem

zz 。、 k I

Velid for erbitrary complex x , end cemplex Iz'0

,

kJ

k=O 00

Z

( 1.~)

(γ)

.k _ (1 _ x)n

k-O

l xl

+1



"'" 2 'COB

nx 乎号_) 2'l COS

n

M "l乎-r) C08

n

'

k 自 O

n

(1.2 7)

Z C)

810 k>

k-O

(

-

2

.s1n 寸r

'

1 约)三。(叫 )ωabEHf2千叶ncoa 忖)，

(1.29)

Z (叫:)…= (-1)口川 in 云)山

(1 列)艺 C) 叫'仰

川z …

(1.~1) 三 (2: )ωs 同ζ

Itt qι

EEEF/

、

,,, nk

XJ

--n

lcx

x

ba

,

2j

oh

-斗 zr-1:叫

-4-

n(x:时，

+-2 ,

l

飞 丁J

幽

e--.

a冒-

句ζ

k 4' ‘‘,,x

阳2

-叶ω) e…)·(-1Mh/川叮 X)) n+ øin(2k + 1)x

(1.~4)

k

o:::

O

… "

十+(n~2)

-2

knK

'

4'

1 飞( -1L( 斗)

』/

t

Jftz 飞

2 (1.~6)

=叮 cJ川in(叶阶1♂(x/2) 毗 n{?)l 岛、飞，，，/飞飞

Ir--飞

阳 2吊

(1. ~5 )

e os

，，‘、

吨'''

飞飞 ZE，，/

B+ 』凰

//飞

tre

，回

ι 』且

、，'!-E

,••

飞/回

-句，

ZJ

、

写J

，，‘、

...

-F吨ζ-

nEF

rtlL

,

k...O a唱··"

) ' ( -- \ x 2k / 2k + 1 k O

-η 一

L C:)

4'

(1 .~8)

1-x n--T ‘‘.,," 1-n '-

-，， E、

〔妇

牛'一

2: (:) k:1.

a，‘、-

(l.m

x-

'

nt 1 n t- 1 (x + 1) - (1 - x) 2(n +1)x

-

'

c::

(1.~9)

P二3

Z

x2k k+1

(2k: 1)

>

斗

-5-

'

r-H+-f x-z

-K

nr1 n+-1 . + (1 - x) - 2 (n + 1)x2

例-T肺、

』/

EZ

?一

11

1 一十

XK 飞、

k

I''s、 t

ω2ω

k-O

.:

(x + 1)

-

z

C:j 叫

‘、.，，

a吨，，

k=O yruedothis::f4;斗。f

飞飞』』 JJ

飞

、‘，，

L斗

L 斗

，，，、‘

勾引

xn

x

二 ('1)k(:::)(kf/γ 三

nz

n

(1

4 ‘‘.,,, ••• --un

‘、，，，，

X - k

form 511/1

n/l

'因…E ，，.‘、

‘

1111/

、

，，.‘、.-

n-K k /III、‘.，，

，，‘‘、

n?川 L

、‘，，

tu『

，，，‘、

....

ZJ

，，‘、

(1. 41)

nk 飞、 11 ，，， r

.K

/111 飞

n?L

L

(

-1 叫 :)t

k = 1

k

ak+ 1

.配 1

The following surn 18 en 5:1/0 or e special instence of an 5:1/1

(1. 46)

L

k==O =

(叫 :)1

1 n(n+1)

2(n+~)

+ 2

n

( 1.47)

k Z 川(:) x+k k-O

-

(;7) x

{η

j-1

j

- (-1)

C!J

zk-O r:7=(":::7· 乙:'，)

,

n

( 1.48)

1

-6-

'

( j 豆豆 n )

1

•

「 fAVJ

IlIJ/

n

、

+k

x

卡ta

nz

x

}J飞 lttz、

17

-r'' 『lk

「飞

L ('~丁

(1. 50 )

j/

11j/

飞

LUτ

ny

什

勺 n /乙

+nn +

/IIltt\ x

/if--\ k x 41k

k "'" 1

al?

，

ff

x

，唱. ，a

』

-， l。

『

、d

\ht J/

na

••

//l\ n4'\11ls/

L

~+Z

a 卡 kr

x

且

b

tlsf/ 、、

tlt

aE‘唱，

/fit 飞

、飞

++

nx

丰1tT

a

-f/'' 111 ‘飞

\、111/

(1.5~)

rIt-、

nZ川

、、.，，

叫/』

EJ

k-J

/IIIE飞E

/ltl飞

…

良/

，，，‘、

AY--

-KX \lII/

j

阳z …由2

where ~也

e

rz

2 1r i/r

=1 (8

=

,

(r-1 ~ a) pOS. or neg. integer)

\ivz

乎 COB~j)n

cos

(0-2;)3 7γ

r

j -

l

川时

/fIL

=0

nu

//I 飞

功另 k

z:::

(UJr j)'a(1 千去电Jr j 户

1

(n ~ 8 主 0 ， r-1 注 8)

Z (1 叫 j)n__= 三 Z( 叫nms平 j .. 1

j ... 1

(156)Z。 {Us 刊 2nHmi号'

-7-

[号.!J

(157)24:1) →F十 2W(?j'

(1 归)Z (;)→ {2n 十川 μ 抖， ( 1δ9)

z C:)

在

k=O

(1. 60 )

k

(1.61)

k)

L( 叫 n : a:

K

(xy)-(x

X

~lC

n 4- 1

n 沪 1 皿

y

a:

'

x - y

O

L C: k)

2

n - 2k

z

L

n 斗 1

k

...

x- y

n

+1

x - y

wh町et

(1. 62)

+ y)

n -

x =

l+F口， y a: 1- ，r;:+可

(叫 ni 丁 (2COSX)n-2K E"::::1)X

k=O

[亏7 ( 1.6~)

Z

(-1) k

t k (":丁

(1.64 )

2

k=O

C

:k)

n

~

2

-= - n

n - k

k 嚣。

[+J

COB x) n - 2k

(2

n

k

here r x

(+)

-= n 2

= 1 +.俨亨丁，

-8-

x . n-1

+

008

y x+y

nx

'

n

y-1- -V耳可

,

(1.65)

L

(-1 )k

22"-2k

C:丁

" -

k-=O

2"

k

n肉ι ，

(2 008 X)

e08(n 千 2)X

'

aE

,

n +-

a唱··

斗

m艺

c=

n

nζ

k

专

‘"

-=

'

n 千 2

(-1)n÷ ，

1 岱~ (叫 k

"+2 -2 "+2

h

ζEL 且 句

b 一+

(

(" ~1)

b且

CF o-B-k

k-=O

(1672]('1)k(n:)

+1

n

A叫一+

nζF

1 担(-1川

(

..

(-1 )叫

O

if' 个'

L ， if 才 n n

'

(Cambridge Msth. Tripos , 19究; Hardy , Pu re Math. , p9ge 445.) /it 飞

\i/

♂+ (_1)"2 n

,

n

(n 杀 1)

k=O

(

1 叫z l

n :

k).k

(2 Z a

n+-1

川

二十

n-x-

-x n

x

,

where I

O

\(1.72)L

(叫 n :

十

(-1)

(2z)

忖 '

n/2

千一门

a::

)

2 户 (2叶 1 -t-{ï工石)" -(地斗 1 圣J丰石)

(1.71罢仁 γ= k

+ 1全J亨丁正

k) 2叫

=n 十 1

k-O

-9-

,

z_

-x E二 ( 1 千 x)2

'

叭 l/

/IIt 飞

ε … (

1 拦 C: 丁=

(1.75

2n

,

( 1+ .J雪)n 十 1.(1-4吉) 2n + 1 -ý歹

E 凰n

E

and

a

n-th Fibonaccinumber , Where aos1·la1' n+1

)艺 (?1)kfi 丁 E

-=

8

+a

n

片~

n-1

+ (-1)

(-1)

[:巳~] ,

2

k-O

(1.76)三 (γ)

- ZC"k-)

k-O

b且

川l/

rjt1、

叶位

∞汇川

c: γ

n

自 2n

x

b且

x

十

只U 吁t

r 飞4JtL

/ftt 飞

+k k \ll/

+xn +

'

k-O

,4 n

k

吨'』

1····\

叭 l/

/,

,

-10-

K X

1/〉飞 J

n 也… 「

n

…

(1.79)L

aa2rEusing (1·7A)

k-O

PZ

auo

+1

-圄-2 2n

'

同ζ

b且

』民

『ζ

，，ttE飞飞·

-

nu

内ζ

E

Z C 1)_ / n

ft1h\ 1llIFf n'-n 内ζ

n

十

吨ζ

=0

、11l』/

b且

,)

(1.8

缸十

k

//-11 、

au

_2n ~

叫ζ

n 乙甲 勺 h Z

au

\III-/ n -n

d

,

(n 注 1)

,

:

竹 il/

+、

叫ζ

nd

n

同ζ

JI--\ n /,

句ζ

且

nz

叭 1I/

n

nt

-L

斗

L

nu

、 /Its-

nγLh

k== 。

-n

n >/

nv

叭 1·j/

nζ

t

吨'』 同ζ

，

、川 l/

句丘

、‘，，

，，‘、

、

-1j/

••• n

/III-飞1

飞

吨ζ

、‘，，

//『t11

b民

=

n

\//

‘•.

,

n -n

n

\111/ nk k

(1 +-v'X)

x

+ 2

,

n-1 -

2

'

肉，』

‘、，，

aE·

，，.‘、

1111JF

ζ

z 乙:)

E

n

2

(n 主 1)

-11-

(1 -吗rx)

n

'

… n

nJ''

nk 、

呵'』吨

,,

/tt 『lt1飞

b且

，，，‘、

4··‘•.

k~O

( 1.89)

nk

肉，』

Z

//I1飞1

(1.88)

E

n -n

O

「lJnv

RU

a唱··

，，‘、

吁VLh

c:

l

良J

(1 附 2 k

呵'』

…

nu ‘‘,,

//ti\

n

斗

叫 ν

/lt·\

-2

'

日斗

I

n飞

1.则~ (_1)k (2:)

(1.91)

艺 C:) -

mz

k-O

(1. 92 )

=:

(1

，.、n

+1) :~' ~ 1)

L Lk: l) n

(-v2 )n C08 芋，

_

2n - 1 -

,

2

(n 兰 1)

k...O

2

1:卜1(C n :) .叫)

十

k

叭 ll/

十

(1.9 4)

n h… n乙 ? 之

、{之主 j)

…

,.. . .、n

zc::)

2n -

2

'

k-O

[吁立]

(-OkfnJ 丁

-= (-1)

2

n

k....O n

n+ .Ji)白- (1 - ...;王)

(195)ZOLklxk'

(

1.96) 三。(斗)k乙k.+ 1)

(1.97)

Z

2-./言

(1+1)n-(1·if -

n

,

(n ~ 1)

sM)n sin 子，

21

L川 -2n-1' a 唱··圃

、

，2、

Enζ

a冒··

-12-

n

..

'- '- ,-

'

，唱，.

‘‘，，圃

幅-

2k+υ

， E、-吨'』

斤 2n 〕 =(-T2

A唱···

叫 ?L

(1.99)

-- n '

k ..，。

(1.100)

Z (;:::)3E "?/μ

(1. 101 )

2n - 2 .. (2n - 1) 2

(n 注~)

)kc:::) 皿 (FJ

,

k 菌 o

( 1. 102)

回(B(x忖

ZN)k(::;)k;1

,

symbolically ,

'

where Bn(x)means B (x) 目 Bernou111 polynomia l. n

n

( 1. 10')

Z C)

k

( 1. 104)

:oc

O

主 C:)

k.也 O

(1. 105)

-t-CJ +

.

k. k!

x k +1

1)

x

k E 飞(1-

(1 - 2k) 2

.z c:) • L Z

-J1万 k x

(11(2:)

-

2k

+1

22k

"'"

k 2

k == 1

'

, (

01) ,

,

n r - 1

(:)

C肖3

(乙 11)

Z

k = 1

z

c=旬-

n

kC: 丁

。 ~ (叫 :1;XK+:?)-ifj;二;二 -24-

GO

唱A

w民

-t:~]

L且

k

l阳μ阳叫?川川川 (臼 剖 222约) Z 三。 (:)C:工: :kJ 护)

飞i'/

x'bh

nζ

f

o

十四

、、 E ，，

Jr11 飞

nζ

/FEE 飞、

nLh 勺

xk ve

er Fb

AUhu

•

内1/

+

x2 n1 Il

内ζ

nnkk

/iE 、

认 ly/

十

，

、飞 K E E J

Jftt 飞

k

n

+

』/

(, .26)

x

… ZC:Y:::) -云rcγnE;二)f 十

(, .25)

X

ffhv

十

、11tIJ

nz

/fi1、

(, .24)

、

n?… /μ

/iu XK x +nnkkQll/ /IIE\

π(x242)'

(, .27)

Z C::)(::DZ;::: 刀:引:(C: 刀丁γγ2 ♂产2缸n~圄 ι 口:峦 峦峦[c扣(仅2川阳 4

(, .28,)

L (:JC:j-t::刀，

(, .29)

z (:)(二 r) -(::刀，

(,.,0)

2(:VDk zn( 叫: -) .

(,., 1) 艺 (-叶叫叫 1η) (,.,2)

L (-1串斗卅j 叫 1ηJ)产叫 )kk丁(:)(n二:二k)卜= (←串斗J1 门)产 2 (ι 斗 )γ1叶川+剌忏川川(←忖川唰斗卅1η) = (:)

~:172;;!!fL)!

'

n 'Ar 1II/

r

、

baH\i/ a -r

，， IE 飞

fftt飞 t

叭 luv

b且

fFE-、 -

…L 勺川

ZJ

!以 5乒引号川; 艺 州 CJ比 汇 L (2nιn 二 n :斗k) -叫川(←归川-斗 1

:~.~切!巧5引肪予另) 三;(斗卅d 叫吨)户叫盯 k气 k仁:)(2' (: 二斗!二 :k) - / E n

人州)

呵ζ

+k+k

，，S ， EE-J''EEE、

4E

U 飞 川

3jill}

2 n n r · K -' n k n 1 2 n . -K -v hn K

ZJZJZJ

rEL

-C2:j ·

2n - 2k

叭1l/

陀ZZZ 二

5110)Z (iyn

2

川 >2: (叫二nk)C叫 :71)E(仇 1) 三

nz … nz

n-

"

k

k- 。

/ftt 、

n

RJnn .1 、 l/

-35-

\112/ nn

zJ

/ff飞 t

飞

11fJ

、

2-5

呵，』 肉，』

+n n

n

吨'』 吨，』

-K

j/

，，Il

叫

b且

n』 J

呵'』

…

k

、、IJ/

L斗

/Itt 飞

，，，ZBEEE、

\飞l/

十­

件冉，』

L

~.116)

/rtt 飞

~ .11 日

+n

nn12

,

内J 』

n』 J

呵ζ

1

飞E /FEEE 、、 ，，/

『， t

十

tte'/

、

飞飞EES''

/'』 E飞

帽 4J

、、 EEE ，，

b且

/''BE 飞

111/ 、

frtI飞 E bι 、‘，， ，咱

a唱··

A嘈

内Y

/Il、·

黯

，，

t11t'''/

、 It--

k/th

川l//

// 飞飞 t， t 吨ζ

飞

飞

吨，』

11'/

Ett1/

/rttt 唱'唱'

!li、

n-k

十斗 吨'』，，，，

/ftl 飞

叫4

叫ζ

肉，』

111J 飞 b皿

/JEE飞 E 、EEEE/

/rl 飞

冉ζ 内ζ

L且

ηl/ b且

飞 、 111''

/il

』且

(R.R.Goldberg)

12公)z (叫 :)(nTm1)n艺1 十但 412\

斗门/

+1

j .. n

唔 (:)2 三卡 (2:) [J~l 才

n

-

hZ

k 嚣 O

, (斗 )n 句4

-36-

1 k -

' k

2 川 nZ川 nZ川 nZ川mz川币

，，‘、

kOLU

+ JFEE--nk

nz 1-k

+k

nk n

k

n

n kj

hl/ nkF2 2

nud-

j

,

1)n , asn (,

j"n

o , nk

‘‘.,,

n

x

… nz … nu

•• ••

nk n -n k /Itt\

x ,,,n44 SIB- x

VVA k44k nk

n

叭l//

mz

。。

126)

Z k-O

00

(:r

-I

2n 十 2k

，仇杀 1)

(:) 2

,

2n - 2k+ 1

k=O

C咱

127)

L

(飞) ~

2 (+lk

(+)

k-O

dx

τod1

k=O

L C:)

1

J

-Jfu1·zE

1 - 2k

'

- z2 91n2 x

dx

,

o

BESSEL POLYNOMIALS.

9)

Yn(x)

=

E

x

‘、.，，

，，‘、

•• k ,-

叫ζ

且

li1/

、

-L

乱圃，

JACOBI

呵，』

x

e

m(-1)n

/,,nk t111ts/ n EK

fF』』』k、、

L

aτ··

=

K 、‘，，，

飞 (x)

k

F 入

，，E‘、

?气。)

L C)C: ,1咐_)

k

n

nd

,

nx -x2

D

e

POLYNOMIAIβz

1)prb) 位 )z

n

+x

hu

，

x

a

E

x

Dnx

, 飞

-;7-

fEftL

KU

十

x

a

十

n'

n ，，‘、

aτ··

2"

、‘，，

，，，‘、

=

(r】:丁 :~:)c~lrk (干Jk

Z

f

J?J

LEGENDRE POLYNOMIALS

Rodri恕les:

Definition by formula of

Jτn:

(3·132)PJN "2n!

(x2 -

1 户，

1n terms of generating functions , c:ao

tn

fnh)E(1-23Et 十 t2 ). 去

白 /μ ?

n=O

ρ

三

n .... 0

10 terms of the Hypergeometric fU nction ,

pn(x)z 附+ 1 ,

-n ,

1; 号~)

1-1eoy Bumme. tions of the form 5:2/0 1n the pn( 仆，曰 nd

mz

the f、011owing table Of equivalent hrmsof pnhj may be

found of use.

、 k11//

-n nζ

呵'』

飞(叫亏~r

n-K k /It-、飞 l/

(~.1~4)

飞\ /，tt

…

(~.1~3)

P (x) '* 2-n n

1iterature involve

n x - 2k

zcl (芒-}-)k k

m::::

O

15)Pn忡 ZC)C:j C; Il

-~8-

z n

(~. e6)

几位) -

(~_ 1)k/2

(:)(2:) 2-k

(

x -

(~ - 1 忖

k==O [才可

(}.1'7J

p_(x) -= n

L ~:)C:) 产 r

k-O

叫~

c:r: ::k)

x

2k srxn 嘀x +

x-

1)

'

(R.P.Ke 1i sky)

/2

/7(

窗户手 l

仇"气 +Jt)n"

'0

Ij/

qζ

f''E、 E \飞lj

，

k - 0

/t 飞

+ L,

nk k 'n

o 十

盯/syo

()问 Pn(x) =

向 1一节

wrp

k

,

钊

k-O

nγLh 万

叫 z (-:X: ~J 产 c (-1川 W x

k

-L

(X + Jx

2 -

1ω 守主)n

,

(valid for integral 乞> n )

-,9-

Integral form of L. SchlAf11:

p _(x) = u

2

(t 2 - 1)n

II

~-~~~ Tr l

2

.n -r 1

D ~.

U

dt

,

(t - x)

c ~here

C 1s a circ1e counterc1ockwis6 about the point t ... x • In genera1 one m8y define p_(x) for a11 rea1 va1ues of n. n

and relations such a8 the

A『··

a 唱，­

.

b斗

，、，，

(

‘•.

,,

(1 - x)

n

+1

fo11o~ing

Z (n:

are found:

1

k

=-(1-x户 L

k-O

+-x-x

T(z)E PZ

then the above reads

f(n)

E

1/

if one defines

。r

飞，

/tart-

坷 -n-1)

-件。-

x

4

c: r

z cr

xk

.k

,

A G即ERALIZATION OF THE

叫 Z X+JEKzr: 丁

VANDE~~ONDE

CONVOLDTION

(吃了>j 二;;二;fγ 才

y

y 十 (n-k)z

(contains (1.125) as special case)

Zc t:

MZCMKYCn寸 -

叫 zc: 丁(Yn-工) ~.145;

kZ)t- 了

t-

- ZC~:: ，

Z (丁才(PJ 才

p

z t'

+ •

1

(Jensen)

t

zk

_..n-p-1 ~ .

I

, (z - 1)

t

0 三~p~n-1

,

IC;

z

n+1

-

1

•

P-=D

,

z - 1 (Jensen)

(}.146)

Z

(X:γ;二

p rqk

(x

t

kZ )(y - kZ)

p(x+y-nz) 才-

x(x "t- y)(y - nz)

k=1

(:)(士刀

Z

15(nz

,

kz)

.1 1时 Cj斗(d

- 1)

k = 1

E 士C)Z kz1

(M-:7

n

(")

C:j · (归H恒8吨问坤时阳叫 咐甲中g萨伊伊e盯n n - 1

n - 1

川7} 艺

nxq

(dk - 2)! (k - 1) f (dk - k )l \ ,

-41-

n-

nZ -

n 十 k

(Vnn der Corput)

,

(Cbu 吨)

(cf. (7.18))

竹叫川

274

·+1

r 飞

Ln/l adttt

， hj 4d'

，，『

+泣)xmmk

叫

、 J

、

111

11l

blu 飞/，‘

，

，

J

rrk

飞

斗

、J

啊M

J

e

x

/lh川

.1 、112JIVE

》Fh，、 -+Y-

XmL

+mtd 飞

田口飞

k

Y­

吁飞 ‘11idr

IIEEE飞

扣儿出

4EWU

飞 II/)(

「

旦

飞

土 23EPJ

17LZ

/ii((

at4t41

〈-nzVHH

e'=-r

州i/

n1)

1l/n

飞

m

ιL4t

-n

-·工

一--n

l/-x 飞

、

1i

/d 川川

11

， -、 -EII J

xa\11/; 俨回

』，/Y-xnx

IrsEEE、=

、

+m/IIEU\il\l

μkah+缸+ /l 飞

-h\二 /

5FHJF

nn

484E1

二

〉民

hζ

11j/

飞

t

/fl飞飞

旦

呻

王h

』

/l

、.，，、‘，，，

J

飞

nn\/μ=-YLPn?L 回

叫-uxjkk /luv/ll飞

·. ·-LA .

,

可J

-乌2-

xi' O(

，、‘，，、‘.，

,,

/lt 飞 n 2

1iJX

、 Jk

rktit-

1

555

ZJZJZJ

..

,-

=飞

/ I l 飞， nk/Il 叽ll'/tt 飞趴

aa

+'BK+m/l

、‘.，、

斗

eAt

Et--v itE1kk1f a t , a··A·· 1 + U X ,""-++-+ . illa-knkk

678

Y

'XUX+n

(… 3川川川… 1巧5刃叨叨3引切) :趴C:川 )C二: k) = 丁士子什钊 斗(-寸丁Xxm - 1) τ

nyLMn?JLMn2Mm2ω

h』 J -'‘、 ，SIZE 飞

ZJZJIJ 寸·叮

F

k k z r k .X+LEEf-F+ < IILL--

-r-mrr fi

4 ••

sns +myn m-JIlt/SEE-++ s ? V · n a + r 2 2 b h U ta h ) Z =\1.h---a -41hu/It--4 -i o x -a 2 2 u · + oa1c AB-4 >

\l/+-3\l/ k k kix xaitEFX-k x1 x x k 2=.-) RJn+h1s , lon+++n

QJD12

3 TO TABLE ADD工 TIONS

叫5川乌):(Jj 儿二) = (x: 丁 (b;? ， m=Irl ω

n

，，/

k 扎

、 EEE

n:

向 d

:(JfjC::)=(J: L(_1)k (:)(γ)

5.160)

JIt-飞

~. 159)

x + n

2

h

.,

(叶 )nω 丁号722a ，

川 61);(-1)k(:〉(nLγz:: =(4)n(nL 丁户，

川62)i ←1)k(:)( n:丁产=川 (n 2:) 产幻，

3163)i(:}( 飞2)zffif: 〉 2n也

(Carlitz)

;;16ui 叫:)( ~2) =川 飞飞

IIIJ

no

an

飞ll/=

--'-

-n

址

nad 2JKK 1L + '

,

BT enn er

、‘，，

J0·K

飞

飞

飞 J

、 -EE

，， r

飞

dk

，，IEEE

Vuqr

r 、飞，

'W(dh

、

、EJ

ζ

，

『\、，/一-­

队lj/=

飞+23

JK

飞

-斗 3-

占JJJhhZY

叫J 』

AI

-x

R

飞 J

xr

Jn}y

、

L且

+『

，、

D』

'''ttt、11

『飞、 /---2

yfIl

xn rt2

-11·LLIll ta kzk nn + + E K n k 2 2 Z+L22 2+florkntlo 2 2 n E 1 2 kx +-11l/2( r k y x2 xk KIBE-rf''SEEt-

『F 』

「 ζnn r'' 飞飞 』EE

「 4

k=O

，，， st‘飞

LM

n勺 、‘，，

冽l

，、 J

rofo

--14

切

pu

/lil飞u

\ll/

在

EEEEt''

q

、

/FEEt飞-

，，，EEE-‘飞

品

ny

'吨J

冽··

rbRJ

mYLMJ Km/IIlt kn, \lj/ + qq) P + + 4Ea ppkn + kmE

(Brenner)

2 xa 、、lj

=(

; k)

rFSE 飞飞

飞 ZEE-，，，

T(;::)XY(「x

，，，，，

)

4·k /'SEEkk x -n xn ++ -K = 飞 EEE

(3川川川 吻 166句9

2k

+ 1 n +k + 1

.(3.168 )

，，SEE-、

BFFL

'

7) '

(3川川呐 1η7mJO (3川川川川 1η171)叭川1忖)

(C; :二:)门(γ) =(:) .

:

(3172)ic::) 仁++:) = (二)

.

…;(丁) (丁) =(丁) . (Graham and Riordan)

(3.17鸟):(气了)(;1)=(:::) ， (当丁75) 二(斗(::7) 产 n

(3.176) k=O n手P

(3.177)

二

k=O

=(:) .

(;;;)(;工) 2盐+1=(:::) (JL:1)(P; 丁 = (γ) 俨P tlMoriarty" , H.T.Davis , et

(3.178)

了

k=O

( 2p : 2k)( p:丁= (3.177)-(3.178)

2ntpr; 丁户

are equiv.to (3.120)-(3. 121).

-乌鸟，

a 工，

:•

(n/2]

k~r

1)k (n \- I J \ k

k\(k 飞 2山=川r{ )~

J

r

飞

n + 1 \ 2r + 1)

(Marcia Ascher)

。[乞 )k -

(companion-piece to (3.179) these are inverse Moriarty formulas -

)(38

a

+

4t

E1tI''

1

k

x+

(Brill)

2b +

b

?」

At

//l

=

a +

飞

、.，，

(2a + 2b + 1)!(2b)! (a + 2b + 1)!b!(a + b)!

，，‘、

= (_1)b

2b +

n

=

飞

AE

,,

/rtt飞 t

。咱』

krLK

、、EE--EJ

kk E·E·、

、.，，、‘，，

，，、，.、

、.，，

4 ••

，，SEEE

n?JL=tyL= ‘‘, IIl-xkt xk ++

t> 2)

2n- 2k-1 ♂( :r)

afK(n : k) ( :)

2b +

a

(A. Brill , Math.Annalen , 36(1890) , 361-370) ere

1飞(勺 c: 丁

/

2t = 3a + 2b + 1

Discussions , proofs , extensions by Gould , Mathematica No. 3 , August 1961)

1切:(叫 :ì 乙 :;1= 川

-鸟5-

Mononga 工 iae

TABLE

.4

SU}.1l1.A TIONS OF THE FORM S 11/1

n ,1)

二 k-

(:) j

...

C:)

(~ )

(n: 1)

1 )

(:十~) n+ 1

e:

x+1 x - z+ 1

n 1. 2)

Z 川 C)(b:k\(b:k)

c n+c

(nu) b - c

n

n I.~)

>

L 川 C)

n-k (xyuZC)(x-KF{1·HK)

ZCIB; (γ)(4;7kzo

|月二(叫:)

k

1

g

Z

k=O 2n + 1 i.7)

Z

k=O

(:) 1:

(n+2x) k+x

k( 缸 k

SO(x) =

-叭l/

1.6)

,

1)

一-Tn

n

,

y 十k

x x+n

2x+ n + 1 (2x + 1)

L-x 的

tnH 十 2') k t- x

-乌 6 嗣

rz

…

E

,

Sr (x) ,

+

/III\

wh1ch 1s (1.42) again.

叭l//

1

-/'『 tt、

-n

s 1(n)E 2(2n

y

(Y:k)

k=O

1.5)

(b ~ c 0) ' (R. Fr1sch)

rk

2

n

20

2k

(2~)

= 2

Z

k

Z

k

k -= 1

1

=:

k

2k

(2: )

=:

1 - 2n

(x 千 1

L (:) (k: r)

n+ r )

-Z(n;7xk

.. C:

r )

n a、

、‘.，，

LH 『

，""可

...

OLh

(2k+1)j1+(-1)n+kj pk(X) I

(n-K+1)2k 十 1(

-乌 7-

n+k 手 1\

2

...

x

n

'

民 l/

IF/-mx

飞

k 1 当)

hn可几

kEO

l :)

'n

k

(_1)k

工

+1

2

1

L

-JrtE、E

4. 12 )

』

n

_1{-1 ( : )

n

F/

、、，，，

，、

BM句·

a2

-T+

n

=2

n

k

l

z

-x

2

hO(2ni1)

aEE' - ..

内ζ

-=

E

飞

÷仁(:)

4.10)

』 1飞

(:) hofn;1)

皿句ζ

~

-tf/-

iI1·

内 ζ

xx a··Et

KEES-、

』

,,

(-1 )k

斗 .9)

x-It

2: c:)

~.8 )

4'n-xaa n-+IMA

Z 川 k-O

rEK

Lt+2}

(川) n 牛 1

2k

-…回5

n 斗 k

(工)

n 寸「

>

刀

4k +1

C:n 十t- k:k)

L.-J

k-O

2D 十 2k

+ 1

2

2k

e:j

'

-

n

Z

川(:)

..

'回

c+ k叶?

k....O

IT h1 s

(2k 斗 1)(

n 千1

'

了 :tf

may be rewritten as an infinite series actue l1 y: 1

二二工 1 牛 1 n 步'

+

.....

(1 吨 )(2-n)

1 4n

(2-电…

2D

7

2

--‘万.

1

1 小· -(2D-1 川'

(H_ F. Sandham)

'l

1 十 2

号(:) h1(x:or; 与

( 2:) R(x) 主 o

1) 艺 (·1)YDK2\- ，川ed 1 叫2:)

'门广 γ?

-65-

TABLE 10 SU阳ATIONS 、

L叶

11

x 斗 2k

』

F/

0

飞JFSSE

1LhXK

OF THE FORM S .11,/0

x

~x

(-2 X )/ arx(-xyz s10

Z (:)\·÷ZOA

2)

,

(7. O

( 12 .1)

1 十 2

-

(气 27)

(丁 γ

valid for

(12.2)

R(x

(:丁(

:) (X:k)e!j

LHt

1 十2

lIIl:

,

,

气

n

-叭lJ

一十

(Y!J

1

-68-

-/'1飞 1

z 二二jin

牛J

n:X-':j

C}f// 、

'言

(, j

一叭11/

n

k

~o C-

-告

2

1

告气 1 十

fh 一十

一

一十

-lll/ E

(:) (:)

(n ~)

>

C:y)

frit飞 -…

xk

飞飞 lF/

a、，，

、‘，，，

，，.、、

Z

+k

y)

27Ilk x-x At/-2

(n ~γ :j

k... 0

(12.5)

/t

k

(12.4)

-、飞飞''』，，，

nOL

•

一户 ll

Z

、11/-k

fri 飞 -n

(12.~)

t

R( 叫 y) 美·告

va 1i d for

nk

'

仁 :D

的 Z

(:Y

2k

CT: +)

+1

•

1 宁 (-1 )

nrk

2 2k.

2

2

(n-k+1

k

..

2D

+1

、飞-tsj

n 勺 /r

4k

1-uh

+1

11:

cγγ2 4k (

This moy be rcwritten in the form n .5)

Z

k-O

(J: 扩

(…) n+k

ε.9)

k=O

+ 1)

.2 4k 4n

(2n 千 2k 斗1)2

(:) C: 丁

y-wJ

T

1

注-k

n

(件k

(X : k)(X + Y + : + n + 1) 1

=

了;丁

Harry Bateman (notes on bin~mia 工 coefficients) ; see Gould , Duke Math.J. , 32(1965) , p.706.

-69-

,

TABLE 16 SU1 (~心a11

(叫: t (

n) (y

/ D1x on)

+:

+

n)

.:;

:)

c: 丁(叫 y: 叫 n)

(x+7+n)(": )

w

(Saa 1s chðtz )

勺乙

~17.~)

+ y + z/2)!

(x 中 y 斗 z)!

for

(17.2)

(x

k-O

Let

(-告)!(z- 告)!(x+;-1)f(z-x +;+1), f r w & 1

(弓J.)!(与1)1

(z- 斗斗! (z -号1)1

仰，y) - (1 - z产叶子 (_1)k ι~ k=O0

(17.4 )

and then it follows that

f(x ,y) _ f(y ,x)

)夺(_1)k _(飞)(-2:) (_x:)

f=i

zk

t27、 (-x叼

~

(1

,

~ 2X)( - X~告)(yγ 号 zk

(* ~ x) (- Xγ- i) (Bailey.)

f (J: )(丁)

?了?

(Th. Clausen)

-71-

、

2

z

'

TABI且 21

SUMMATIONS OF THE FORM

L崎

-BEES

f

』

，

飞

n

ZJnJ

n

、

tltE

A 咽··

、、，，

，，a‘、

E

n

/IJIt 飞

ZJ

，，

b且

n -n

冉'』

/It--\ ZJ

、、11'/

、11i/

(_1)k

ISEEEE 飞

Z

，

(21.1)

ZJ

nK

ZJ

s.6/0

EIBEE nn

,/

'

2』 内，，

TABLE22 SU阳~TIONS

au /it--L

(n 十 b)! (n +c)!

(n 寺 a -f c)!

(d 手- 2n)!

-呵'』

•

+8 )t

+a

l/ 一叭

EA-

(2a) f (2b)! (2c)! (n + d)! (2的! (n 州 (n

+k k/ 4·-+K 7f 飞

+

bι

ban

\飞iy

/Iluven 冉 . ζ

+

\飞i/

呵，』.

bn

同ω i

J

LM

』

+

飞飞 11

- (-1)

an

/JI 飞

E

肉，』.

rrftl 飞 E

Z时

/'』 EE、

(22.1 )

\11/ hk

OF THE FORM S 15/1

手 b)!

a{ b! c!

(n+-b 手 c)!

'

(a 令 b>! (a + c )f (b +c)! where

d _ a

+b +c

By use of the 1dentity

(γ丁

ck)

(γ)(2:::丁 and the reøult 1s tabulated under that headin ,.

hz

Special C8ses

+k fIiL RJ

ZJ

nn EE

，/

aτ··

E

，，‘、

-73-

/ttttk zJ

n

、飞

bh

、，，

可iUW

J\j

EJ hk flttL n

、飞EEF/

/'E·E 、·

冉ζ

呵ζ

…

bEm

(知)! (2n)! (刑， nf

'

TABLE2~

SU Mlt.ATIONS . OF TIÆ FORM

n

-K-'由 y­

且

-7 乌-

飞川一刻!/

、-EE--x

fttE 飞-中'n

11itJ' n-· 、 Z2

IIl 飞

\飞 --J一­

(2~.1) )~

vd··--U

vvn 一-T

飞 /I飞一 /tlh

xk-x YLEvek x--||i

,

S 4/2

(2:)( 2:) (x: 7) Cx : 才(气 -j (Follows from(17δ) .)

TABLE 2 1占

SUMMATIONS OF THE FORM

国

十 27/7，1(·1)

kC)

S :~I:当

( :) ( :)

(γ)C 门口 7

zy

，，，.-、，， -x

''一

一斗

R( x 十 y+z)>

飞，

ve lid for

4'一

(-+ z-rk x-xvw-vw x-zy-rtz-z y

C:Y) c:)

==

f· 一十

(x+; 才

-1

, (Doúg e. ll)

(The 8uthor shows in e pener ewaiti 吨 publication ， that this reletion implies FjeldEtad's relation - (6.1). )

T

-75-

TABLE ~1

.1 )

何?/]

(~1

SUMY.ATIONS OF THE FORM S .4;'

( _1)k

c: 可 +k- 〉(;)(;)(;)

k-O

k

(x 十 c>! (y -卡 c)!

J

(z + c)!

C+: 才c γ 丁C+:+j (x 和 y +z 手 c)!

R"

(y 十 Z 十 c)! (z 啡-x +c)! (x+-y 千 C)f

ve11d for

-+- y

cf +z 千 c)

.,,·-

飞，，­

，，E‘、，

，.四

(Beiley)

-76-

, 1·

‘‘.. z-a'' x-ve--z -x -vd· ,‘‘., B

、

-x

二，，，

--Ft

X

-

> -1

r 飞

:~1.2)

jfcx- 1: '/2)( :)(- :)

R(x

(DougeU)

'

TABLE 71 su阳ATIONS OF THE FORM

sI6乃

One of the m08t gener81 identit1es known 1s th8t of Dougall , which m8y be expressed 88 8n

516/5

or 88 B 7乌

"ith unit 盯&Um8n也·

「「门盯+丁丁::-丁刀咒)川川川 (:ο:)(:)( :)(:) (叶 y γ +刊仰山 肌 2ce川手

(71.1) 艺(衬

y

口+丁;7+ 丁 γ 丁叹飞γ7川忖("叶+γO ("t 叽丁 (X+ Y 飞 ?+;7+忖0+)

C+γJC寸+j(叫 γ 丁 C:j

,

了:+ n) (Y+:+丁了?丁(叫 :+c寸 TABLE 97

Expreesed 88 e

7气。 ne

hBS the equivalent form of Dougall's formulB

「息， 1 -1- 8/2 , b. C t d t e ,

(97.1)

7'6

..

1I a/2 ,

-

n ;

1 +8 斗， 1;-息 -c ， 1+a-d , 1 +8 吨，

• aHC4)n

(1+ak(1+a-b-e)n

(1+8·bd >n(1

(1+a-bh(1+a , ck

( 1 中amdhn十 a-b-c-d)n

where 8nd

1 + 28 .. b 乎 e 千 d + e - n

(x +n - 1)!

户-t- n - 1)

(x)n-r ，-唱、~- n!l. -飞

-77-

J

n

i

' ,

n... 0 , 1 ， 2 ，当，..

TABLE X SU肌ATIONS OF THE FORM

S.p/O

An Bsymptotlc formulal n

2 (:r

pn 2

'‘~

-VP

k-=O

p -1

侈;-)

2

a8 n -;..

00.

for p _ lnteger 二~ 1 • (Polyp Bnd Szeg~.)

n

Z C:)

x

k-O

~)

k f ?xP ?- L jp - kr x.- ~ x (k+1)p ( ) ln+

-

(n+1)!P x

(n+1)p

'

(Gould)

t. L~:-l) L'-l)j C

p ;:)

C+ r: jγ

- C)

'

VB l1 d for integral r ~ 1 , P~ 1.

4) Z

Fa

nr

+

-(1;γ ，

r n· -r P

---EES Its-aEEL

,

J

E

a咽，

b晶

叭l/

b且

/Ittt飞

，，‘、

,5)

问Z …

r

(斗 )kLγ)(…tyfr· 〉

'

(The threereletlon8 immedietely ebove ere connected "ith very generel eXpansions of Worpitzky , e8 shown by Cerlitz , end were 'rediscovered' by

veriωs modern writers , in perti叫 er by Shen汩的.)

-78-

Z (:)

飞(q) -

Defini tion:

(X .6)

(Stever)

nzvhzoq

(X .7)

(5tever)

KP

Theo the following formulas are knownl

(X.8)

5_(~)

一2

5 _ J, ( )) 口，""

E

n

zJ

qu

n

'1··

= --.!!当 (n + 2

2 S

n

n-I

A

~(~) = ~二5_(~) -二~ 6 n-' . 6

n ，~-"

(n -

(X .1~)

ZJ‘‘,,,

4··

1L s~ '~(~)

民

(X .12)

qu

.,

ZJ

、、.，，

RJ

ZJ

，，E‘、

n .,

、‘，，，

qu

, (X .11 )

2~

5n ， 2(~) ...，一卜 50(~) 十

，，，、‘

(X.10)

(~) --= , 1'.....'

，，，‘、

(X .9)

n

n2 5 n'" n(~) ....- (7n ,...2 - 7n

+ -, 2) Sn_1(~) + -n-1

~) s n-1~(刃， (staver)

8(n - 1)2

Sn-2(当)

,

(Franel) f~r8t

This relation just giveo Wes

proved by J. Frane1 , 1n answer to a

que8tion posed by C. A. Leieant in the first volume (189均 of the journe1

/..

.

..

.. /

.!'Intermédiaire 但旦 Mathematicien8.

Freoel a180 found the

followiog formula es well a8 other interesting resultsI r

(X .14)

J

SJA)s2(2n-1)(5n2-5n+1)s

十 (40 - ~)( 4n -问 (An-5)S (X .15)

JA)

JA) (Franel / St~ver)

主 (:r 小言:c:1; (寸(叫 -79-

KTU 归

-K4d 、飞、EEE

，，， aEE飞、 J

+ 1)

c(q:

(Nenjundiah)

J

，

n ,k

、飞ht1/'

IF- 飞

C( q.

where:

"K

'his genorel result includcs several speciel ceses in the prcceding Ibles.)

Thus: ♂

‘•.

aτ··

，，，‘、

5·K

飞飞，，

k

(n ;, 1)

(0)

,,

(:J

且

、

ff

』

z

If-1、、

向ζ

FM

、‘，，，

11t

，，，、

飞

nub

回

fft1

‘、，，，

户V

，，‘、.

,E··

E

- (-1)

11if/ nK

C(当) =C) 了:丁

阳c 喇 蜘圳胁灿 M 时盹巾 8 加 h10n

owever this 1s too

εxtc 盯nsive

b2k (1 :1 b k) ( 2 :2 ) 1

a

b k llif +m

r

nu

r

』，

飞 tBIll-

a krr rbc 、飞 '1

rtt『t、

•

-

r

=

0

==

0,

J

，

、 //tIll-

‘飞'he''

k11Et--/ a k22 R』 J -bc

rttIFfi、 tt

，

114

alLUFUW

，，，， EE飞

、

1lij/

when

、

I (叫:)

I1 、

，，

/Il--\ a kt4l

(X 丁 7)

JIt-、-

x.16) 主川( :)

to list in the present tables.

b 1 c 1 + b 2 c 2 +… +b r c r , Jhrsber. Deutsch. Meth.Vereinigung , V01. ~7(1928) ， 2;1-4.

5. T..J. 1'e. Bromw1ch , Introduction to the theory of inf1nite ser1es , 2 nd Ed. , London , 1949. 6. L. Car1itz , On a c1ass of finite sums , Amer. Math. Month1y , V01. ;7(19;0) ，的2-479. Note on a formu1e of Sz11y , Scripte Y. athematica ,

7.

Vo 1. 18(1952) , 249-25;.

,

8.

Note on

B

paper of Shanks , Amer. J..1 eth. Month1y ,

V01. 59( 19究)， 2;9-24 1.

,

9.

Note on a formu1e of Grosswald , Amer. Meth. Month1y ,

Vo 1. 60(195;) , p. 181.

,

10.

Eu1erien numbers and p01ynom1els , Math , Mag. , V01.;2(1959) ,

247-2 臼.

11. A. L. Cauchy , Exercices de 1~eth{mat1ques ， first year , 1826 , p.5; , or see c011. works , ser. 2 , V01.6 , pp.

-93-

72-7~.

12. Chrystal ,

A Textbook of Algebra , Reprint by Chelsea Publ.Co. , N.Y.1959.

15. K. L. Chung , problem nr. 4211 , Amer. Math. Monthly , 1946 , p. }97. 14. E. H. Clarke , problem nr.. '996 , Amer. M~th. Monthly , 19岛 1 ， p. }41. 15. J. G. Van der Corput , Over eenige identiteiten , Nieuw Archiefvoor Wiskunde , ser.2 , Vo l. 17( 19'1) , 2-27. 16. Reed Dewson , Binomial Formulery , menu&cript , Washington , D.C. , 1957. 17. Devid Dickinson , On sums involving binomiel coeff1cients , Amer. Math. Monthly , Vo l. 57(1950) , 82-86.

Cf.

Math.Reviews ， 11(1950) ，每88.

18. A. C. Dixon , On the sum of the cubes of the coefficients in e

e

certein expansion by the b1nomiel theorem , Messenger of Meth. , Vol. 20(1S91) , 79-80. 19. N. R. C. Dockerey , An extension of Ven der

l~ond'

s theorem end some

ep?lications , The Meth. Gezette , Vol. 17(19") , 26-'5. 20. John Dougell , On Vandermonde's theorem , end some more general expensions , Proc. Edinburgh Meth. Soc. , Vol. 25 , (1906)

,

114-1}2.

21. H.. J. A. Du parc , C.G.Lekkerkerker , end W. Peremens , An elementary proof of a formula of Jensen ,

see Meth. Reviews , Vol. 协( 195}) , p. 858.

22.< Wi l1 iam Fe l1 er , An introduction to probebility theory end its app l1 cations , N.Y. ,

19~O.

2'. J. E. Fjeldsted , A generelization of Dixon's formula , Meth. Scendinavica , Vol. 2(1954) , 46-48. IKJ-Franeh a note published in 1'IntermAMire des BEatt16Laticiene , Vol. 1( 189町， 45-47. ~5.

~6.

,

en

G1ves sums of powers of binomiel coefficients.

ex~en8ion

of ebove , in vo l. 2(1895) , "-}5.

R. Frisch , Sur les semi-inyerients , etc. Videmskeps-Akedemiet s Skr1fter , II , No.} , Oslo , 1926. Q.uoted in Netto' e Lehrbuch , by Th. Skolem , p.}'7. -9 每-

27.

Jekuthiel Glnsburg , Note on Stirling's numbers , Amer. Meth. J.:onthly

,

Vo 1.

~5(1928) ，

77- '3 0.

28.Karl Goldberg , M. Newman , and Emilie Heynsworth , Combinatorial Anelysis , Netional Bureeu of Standerds manu6cript peper , NBS156 , 1959. 29. R. R. Goldberg , Problem nr. 4805 , Amer. Meth. Monthly ,"V o1. 65(1958) , p.6~，. ;0. H. 'tl. Gould , Note on e peper of Grosswald , Amer.

J~ath.

Monthly ,

Vol. 61(1954) , 251-25~.

, Some

;1.

generelizetions of Vanderrnonde's convolution ,

Amer. Msth. Monthly , Vol.

,

;2. }.~onthly ，

6~(1956) ， 8斗 -9 1.

Final enalysis of Vandermonde's ccnvolution , Amer. Math.

Vo 1. 64( 1957), 409-415.

;;.

, A theorem concerning the Bernstein polynomials , Math. Ma e;azine , Vol. ;1( 1958) , 259-264.

,

~4.

Note on

5

pepcr of Sparre Andersen , Mathematice Scendinevica ,

Vo 1. 6( 19 5'匀， 226-2;0.

,

~5.

Dixon's seriee expressed ae e convolution , Nordisk

Matemetisk Tidskrift , Vol. 7(1959) ,

7予76.

;;6.

,

Note on e paper of Steinberg , Math. Mag. ,

;7.

,

Stirling number representetion probleme ,

Proe. Amer.

;8.

l-~eth.

, ,

,

11 (1960) ,

1959) ,46-48 •

鸟也7- 乌51.

Generalization of e theorem of Jensen concerning

convolutione , ;9.

Soc.

Vol. 刃(

队Jke

Math. Jour. 27 (丁 960) ， 71-76.

EvaluetloD Qf the finite hypergeσmetric serles F(-n ，告， j

+ 1;句，

to be pub I1 shed. 与o.

,

Fjeldstad's series and the q-analog of a series of Dougall ,

Math. Scand. , 8(1960) , 198-200. -95-

55. T. Narayana Moorty , Some summationformuleefor binomiel coefficien tB, l.~ath.

Gazette , Vol.39( 1955) ,

122 幽 125.

56. C. Moreeu , Another so1ution to the

川。blem

of Frenel in 1'Intermldiaire

des Meth{maticiens ,Vo1.1 , p.47. 57. Leo

l-'l

oser , prob1em Nr. E799 , Amer. Y. eth. .t-1onthly , 1948 , p 、 30.

58. T. S. Nenjundieh , On e formu1e of A. C. Dixon , Proe. Amer. Meth. Soc. , V01. 9(1 9,58) , 308-311.

,

59.

/

Remerk on e note of P. Turén , Amer. l.:eth. Monthly ,

Vol. 65(1958) , p.354 •

60. Eugen Netto , Lehrbuch der Combioetorik , Leinzig , 1901 ,

~ec.Ed.1927.

Reprinted by Chelsea Publ.Co. , New York , N.Y~ ， 1958. 61. N. Nielsen , Hendbuch der Theorie der Gemmef、unktion ， Leipzig , 1906.

_,

62.

Treit: éí6mcnteire des n Olrbres de Bernoulli , Peris , 1924 , note p.26.

63. N. E. N巴 r 1und , Yor 1e sung ,: n_ ð b~r Dif.fere.~ze，n reçþQ.u ng ，.. >>~r.l1 n ， 1924. Reprin~ed-by ChelseaPUbl. Co. , New York , N.Y.; 丁 95 件. 64. Q. P61ye end G. $zeg巴， Aufeeben undLehrs8tze , Berlin , 1925. Note Vol. 1 , pert 2 , problems 206-218. 65. John Riorden , An introduction tocombinetoriel enelysis , N.Y. , 1958.

fr6•• R咿 1 ，

Eine

ber口的enswerthe Ide川 t l! t ， 问pe'e Archiv , (2) Vo 1. 10 , 110-111.

See review io Fortschritte , Vol.2; , p.262. 67. Heinrich August Rothe , Formulee de eerierum reversione , etc. , Leipzig , 179~. 白.

H. F. Sandhem , verious prob1ems , Nr. 4519 , etc. , Amer. Math. Monthly.

69. L. Schlßfli , ErgAnzung der Abh. ðber die Entwickl. des Produkts...etc. , Jour. r6ine u. engew. 70. O. X. Schl&milch.

,Vol. a『

7

.•

-

l-1 eth.

, Vo l. 67(1867), pp. 179-182.

Compendium der

h 色 heren

Anelysis , Breunschweig , 1866 ,

II , pp. 26-31.

,

Recherches sur les eo~fficients des fecult6s enalytiques ,

Jour. reine u. angew.

Jv:eth. ，飞TOl.44(1852) ，

;44-'55.

'2. 1. J. Schwatt , Introduction to the operetions with series , Univ. of Pe. Press , 1924.

Reprinted 1962 by Chelsea Pub工. Co. , New York , N.Y. -97-

7~.

B. Sollertinsky , e summ8tion equiv81ent to

Chur唱 's

pro !, osed 8S Question Nr. 12 , l' Intermédiaire des J

problem ,

J.~athematiciens ，

VoL 1( 189句， page1. 74. Tor B. Stever ,

om

summasjon ev poteneer ev binomielkoeffieienten ,

Norek Matematisk Tidsskrift , 29.Krgeng ， 19饵， 97-10~. 75. B. A. SWinden , Note 2299 , on notes 202; end 209句， lv:ath. Gazctte , Vol.

~6(1952) ，

p. 277.

76. K. v. SZily , see review in Fortschritte , VoL 25(1 ,S9}) , 405-6. 77. P. Terdy , Sopra alcune formole reletiva ai coefficienti binomiali , Journel of Betteglini , Vol.

~(1~6刃， 1-:~.

78. C. ven Ebbcnhorst Tengbergen , ðber die IdentitAten...etc. , Nieuw Arch. Wiskde. , Vol. Hl( 19~4)

，

1-7.

79. G. Torelli , Qualche formola relativa all'interpret a. zione

fattoriale

delle potenze , Batt. Jour. Vol.; ,; 178-182. 80. D. C. Wong , various problems in Amer. Math. Monthlj. 81. J. Worpitzky ,

Studien 咀 ber

die Bernoullischen und Eulerscher> Zahlen ,

Jour. reino u. angew. l-lath. , VoL 94(18ô~) ， 20~-2~2. 82. prob.

~~77 ，

Amer.

~eth.

Monthly , 1929.

8~. prob. 句51 ， ibid. , 19另， p.432. see solution , ibid.Vol.61(19功)， 65~ -4.

-98-

• ERRATA SHEETS FOR THEORIGINAL

OF THE TABLES -

PRINT工NG

page numbers refer to original pagination

ERRATA ,.;5 of' ma1ntableø ，如d 11ne up '-5 , 1n{136) , on right read ). 8 , 10

,. 9 ,

(1δ8) ，

read

n 1T

z

1n

,

, .17 ,

1n (2.16) , on r1ght

~. 18

,

instead o f'

+ x)2

,

read

1nstead of'

.

k'

(:")… ...l旦L

一 k

•

•

or(:n)

n~

. _. n

1n ('.72) , read

+ x)2

.

"注 2.

p. 21 , 1n ('.2') , on r1ght hand s1de read p. 27

_x2

z_

(1

1n (1. 118) , addcondit10n that

1n (2.24) , read

•

C": 1) …叮

(1 川….d

~. 14.1

n τ-

-x

.膨

•

instead oT1·(-1)n.

1nstead 01'

(1

,

bottom , read [72J 1n阳ad of 72

t+( , 1)n

T

in (1.71) , read

f'r om

2

1nstead of n

2.

二4in.

1nstead of

2

'0 , 1n (;5 .95) ,

p.

40 , (4.10) , tbe sum shou1d be

read

p. 40 ，如 (4.1') ,read p. 50

,

1n (7.11) ,

l'

(x + 1)

n+ r

1nstead 01'

I

ead(

x(x 午 1). • .(x 牛 n)

nLh 宁

p.

r" :

n

1nstead o1'{

飞 2n I

(:) 1)

.

r.

1

、 x/

p. 7咱， 11ne ;5, read (Z.12) 1natead of' (Z.1').

p. 79

,

change statement a 1'ter (Z.'5) to read ITh1s 1s the genera1 cas8 f'or (Z.,, ).I

,.80 ,

1n (Z.'9) , on left read

). 80 , 1n 1.

(z.48) ,

.J1f (2n>!

on r1ght read γ(

81 , 1n (Z .52) , read

1nstead o f'

/2 " ,

k

J

.n.

1nstead 01' -99-

1nstead 01' kk

.

(了)

.

ERRATA - Combinatorial Identities by H. W. Gould

page 6 formu叫1.46)

1 24(n+3)

for

气~ 7)

叮 ~4)

{z(J+3}+h

page 18 formula (2.24儿 line 2 α3

α2

n

page 21

~

L

read n

~

I

page 35 , in third for

d土 splayed

.n t.. ---n

read

nJι

nJ』 n』 J

:f ormula (3.22) in le:f t-most sum p k -2k for 2- L.- read • formula from top , t.n .-=0

-100-

n品

d

Er rata f' or:

G∞1d ，

page 4，件 (1.31).

"Combinatoria1 工 dentities ，

Rep1ace Rep1ace

(_l)A

si♂(~)

by by

(-1)

k-1

Multip1y the right member by

page 26 , eq.(3.58).

Multip1y the right member by

严吕e 31, eq. (3.工03). page 43

,

eq.(6.5).

Eepkce

卫讪ce

2nfby ♂-k

叫叫2) by 主[1 + (-l)nJ

page

44, e吐·(6.8).Dehte(4)k

page

44, eq.(6. 工0).

工nsert (_l)k i的。 the s urom.ar也

page 47 , eq.(6.35).

Mu工tip1y the second sum by 22n

page 51, eq.(7. 工9) •

Rep1ace (且-x:y-1) n

by

(n+x:y-1)

page 66，叫. (31. 工).

Rep1ace

by

(x + c)!

(x + y)!

---compi1ed by E1don R. Hansen , August , 1969.

-10 丁-

n-n』 f

page 26 , eq.(3.57).

间可♂

page 13 , eq.(1.106).

si♂(言)

"

FURTHER ERRATA

taL

p.7 , in (1. 切， addzwhere p.14.1

,

,

f&-

read (k +

for

in (7. 丁 8) ， the sum in (7.17)

,

1)

Cf. Han5en'5 form this correction.

。f

read

dk

k=1

= -1

for n

= O.

工 ndicated sum correct

for p. 5 丁

A 、 k-1

(2n~竹.

read n -BJO --、飞 /〉 r---

p.51

1)~-'

2τri/r

l1 2n+2x飞 ~2n"""^") ,

p.38 , in (3.1 斗7) ,

e

k-1

in (1.117) , for (x +

p.26 , in (3.57) , for

z

n ~ 1.

in left-hand sum , insert the factor

A 工 50 ， for n~ m

1 k

.

m 共 n.

read

p. 65 , for (2，.阳 )iead (24.1). p.86 , item #51

,

for 'Todskrift' read 'Tidskrift'.

p. 87 , item #60 , add ref. to Second ed土 tion ，. 1927. p.3 ,

introduction ，工 ine 丁 0 ，

p. 69 , in (x.13)

,

for 'indes' read 'index'.

for (n_1)3

read

(n_1)2.

p. 14.2 , in def. stated with (丁 .127) ， deletethe factor

H. W.

Gou工 d

1 May , 1972

-102-

(_1)k.

Research and Pub1ications. selected 1ist:

H. W. Gou1d

1. Note on a paper of Grosswa1d , Amer. Math. Month1y , 251-253.

61(195 斗)

2. Some genera1土 zations of Vandermonde's Month1y , 63(1956) , 8 鸟-91.

Amer. Math.

convo 工ution ，

,

3. The Stir1ing numbers , and genera1ized d 土 fference expansions , University of Virginia Master's Thesis , June 1956. iii+66pp. 4. Fina1 ana1ysis of Vandermonde's convo1ution , Am er..Math. Nonth1y , 6 斗 (1957) ，鸟09-415. 5. A theorem concerning the Bernstein po1ynomia1s , Math. Magazine , 31(1958) , 259-264. ~.

Note on a paper of Sparre Andersen , Math. Scand. , 6(1958) , 226230.

7.

D土xon's

series expressed as a convo1ution , Nordisk Mat. Tidsk. 7(1959) , 73-76.

,

8. Note on a paper of Steinberg , Math. Magazine , 33(1959) ，斗 6- 件 8. 9.

Combinator土a1 Identities , a standardi~ed set of taö1es ~isting 500 binomia1 coefficient summations , Morgantown , W.Va. 1959-60. 100pp.

10. Genera1ization of a theorem of Jensen concerning DukeMath. J. , 27(1960) , 71-76. 11. Stir1ing number 丁 1(1960) ，

representat 土。 n

斗斗7- 鸟51.

conv 。工utions ，

prob1ems , Proc. Amer. Math. Soc.

,

12. The Lagrange 土 nterpo1ation formu 工a and Stir1ing numbers , Proc. Amer. Math. Soc. , 11(1960) ，也21- 斗25. 13. A q-binomia 工 coeff 土 cient series transform , Bu11. Amer.' Math. Soc. 66(1960) , 383-387. 1 鸟.

Fje 工 dstad's

Math. Scand.

,

series and the q-ana1og of a series of Douga1工， , 8(1960) , 198-200.

气 5.

The q-series genera1iaation of a formu1a of Sparre Andersen , Math. Scand. , 9(1961 ) , 90-9 乌.

才 6.

Note on a paper of K1amkin concerning Stir1ing numbers , Amer. Þ1ath. Month1y , 68( 丁 961 )，鸟 77- 斗79.

17. A series identity for find 土ng J. , 28(1961) , 193-202.

conv。工ution

ident 土 ties ，

Du ke Math.

18. The q-Stir1ing numbers of first and second kinds , Duke Math. J. 28(1961) , 281-289.

,

19. Some re1ations invo1ving the finite harmonic series , Math. Mag. 3 乌 (1961) ， 317- 321 •

,

20. A genera 工 ization of a prob1em of L. Lorch and L. Moser , Canadian Math. Bu11. ，斗 (1961) ， 303-305. 21. A binomia1 identity of Greenwood and G1eason , Math. Student , 29(1961) , 53-57. -103-

22. (with A.T.Hopper) Operational formulas connected with two generalizations of Hermite p 。工ynomials ， Duke Math. J. , 29(1962) , 5 丁 -63.

23. Congruences involving sums of binomial coeffic 立 ents and a formula of Jensen , Am er. Math. Month工y ， 69(1962) ，鸟00-402. 24. A new conv。工ut 立。n formula and some new orthogonal relations for 主 nversion of series , Duke Math. J. , 29(1962) , 393- 斗04. 25. Generalization of an integral formula of Bateman , Duke Math. J.

,

29( 才 962) ，鸟75- 斗 79.

26. Theory of binomial sums , Proc. W. Va. Acad. Sci. , 3乌 (1962) ， 158-162 (196~). 27. Note on two binomial coefficient sums found by Math. Stat. , 3斗 (1963) ， 333-335.

R土。 rdan ，

Ann.

28. The q-analogue of a formula of Szily , Scripta Mathematica , 26(1961) , 155-157 (1963). 29. Generating functions for products of powers of Lucas numbers ，丁 (1963) ， No.2 , 1-16.

F土 bonacc 主

and

30. Solution to Prob 工 em 470 , Math. Mag. , 36(1963) , 206. 31. Solut 土。 n to Problem 505 , Math. Mag. , 36(1963) , 267-268. 32. Sums of logarithms of binomial 7 1( 196件)， 55-58. 33. The functional operator Tf(x) Math. Mag. , 37(196 斗)， 38 蛐斗6.

coeffic 土 ents ，

=

Am er. Math. Monthly ,

f(x+a)f(x+b) - f(x)f(x+a+b)

3斗. Solution to Problem 521 , Math. Mag. ,

37(196 件)，

,

61.

35. (with L. Carlitz) Bracket function congruences for coefficients , Math. Mag. , 37(196 斗)， 91-93.

b 土 nomial

36. A new series transform with applications to Bessel , Legendre , and Tchebycheff polynomials , Duke Math. J. , 31(196 乌)， 325-33 乌37. The operator (ax~)n and Stirling numbers of the f 让 st kind , Amer. Math. Monthly , 71(196 鸟)， 850-858. 38. Problem 5231 , Am er. Math. Monthly , 71(196 鸟儿 923. See SG 工 utions ， ibid. , 72(1965) , 921-923. 39. Binomial coeffic 土 ents ， the bracket function , and cornpositions with relatively prime summands , Fibonacc 土 Quarterly ， 2(196 件) , 2 斗 1-260.

40. The construction of" orthogona 工 and quasi-orthogona 工 number sets , 72(1965) , 591-602. 41. An identity involv土ng Stirling numbers , Ann. 工 nst. Stat. Math. 17( 丁 965) ， 265-269. 斗 2.

A variant of Pascal's triangle , Fibonacc 主 Quarterly ， 3( 丁 965) , 257-271. Errata ，鸟 (1966) ， 62.

•

-10 1

,

43. Inverse series re1ations and other expansions invo1ving Humbert po1ynomia1s , Duke Ma th. J. , 32(1965) , 697-711. 44. Note on recurrence re1ations for Stir工 ing numbers , Pub 工.工nst. Math. (Beograd) , N.S. , 6(20)(1966) , 115-119. 句.

Cwi th J. Ka

summatioJ丑lS ， J. Combinator 土a1 Theory , 1(1966) , 233-2乌7. Errata , ibid. , 12( 丁 972) ， 309-310.

乌6. Note on a q-series transform , J. Combinatoria1 Theory , 1(1966) ,

397-399. 斗7.

(with C. A. Church , Jr.) Lattice point solution of the genera1ised of Terquem and an extension of Fibonacci numbers , Fibonacci Quarter1y , 5(1967) , 59-68. prob 工 em

鸟8. 斗9.

Comment on Prob1em 602 , Math. Mag. ，斗。 (1967) ， 107 皿 108. (with Maurice Nachover) Prob1em E 1975 , Am er. Math. Month1y , See solut 土 ons ， ibid. , 75(1968) , 682-683.

7 件 (1967) ，斗37- 件 38.

50. Note on a combinator 土a1 identity in the theory of graphs , Fibonacci Quart. , 5( 丁 967) ， 2 乌7-250.

bi-c 。工。 red

51. The bracket function , q-binomia1 coeff土 cients ， and some new St 土 r1ing number formu1as , Fibonacci Quart. , 5(1967) , No.5 ，斗01423. Errata , ibid. , 6(1968) , 59. ~2.

(with Frank1in C. Sm土 th) Prob1em 5612 , Am er. Ma th. Month 工y ， 75(1968) , 791. See solution , ib 土 d. ， 76(1969) , 705-707.

53. Prob1em H-1 乌 2 ， Fibonacci Quart. ibid. , 8(1970) , 275-276.

,

6(1968)

54. Note on a paper of Pau 工 Byrd ， and a Fibonacci Quart. , 6(1968) , 318-321. 55. Note on two binomia1 coefficient Math. Physics , 10(1969) ，斗 9-50.

,

No. 斗，

s 。工 ution

ident 立 ties

252. See solution ,

of prob1em P-3 , of

Rosenbaw且

J.

56. The bracket function and Fonten{-Ward generalized binomia1 coefficients with app1ication to Fibonomia1 coefficients , Fibonacci Quart. , 7(1969) , No.1 , 23- 乌0 ， 55. 57. Power sum identities for arbitrary symmetric arrays , SIAM J. App1ied Math. , 17(19~9) ， 307-31 6. 58. Invariance and other simp1e techniques as a genesis of binomia1 土 dent 土 ties ， Nieuw Archief voor W土 skunde ， (3) 丁 7( 1969) , 21 乌-217. 59. Remarks on a paper of Srivastava , J. Math. Physics , 12(1971) , 1378- 丁 379.

60. Comment on a paper of Chiang , J. Math. Physics , 12(1971) , 17 也3. 61. Reviews #2033 - 203件， Math. Reviews , 38(1969). 62. Noch einma1 die Stir1ingschen Zah1en , Bemerkungen zu Noten von Johannes Thomas und F. Klein-Barmen , Jber. Deutsch. Math.-Verein. 73(1971) , 1 件9-152. -105-

,

63. Solut 土。 n of Problem 783 , Math. Mag.

,

鸟斗 (1971) ，

289-291.

64. Equal products of generalized binomia 工 coefficients ， Fibonacc 土 Quart. , 9(1971) , No. 鸟， 337-3 斗 6. 。二.

Explicit formulas for Bernoulli numbers , Amer. Math. Monthly , 79(1972) , 斗斗-51.

5.6. Solution of Problem 70-21 , SIAM

Rev 土 ew ，

1 乌 (1972) ，

166-169.

67. A new symmetr 土 ca1 combinator土a 工土 dent 土 ty ， J. Comb. Theory , 68. The case of the strange binomia1 identities of Professor Moriarty , Fibonacci Quart. , 69. Some combinatoria1 identit 土 es of Bruckman - a systematic treatment with re1ation to the older 1iterature , Fibonacci Quart. , 70. Two remarkab1e extensions of the Leibniz differentiation formu1a

_4 --

A new primality criterion of Mann and Shanks and its a theorem of Hermite , Fib 。且 acci Quart. ,

re 工 ation

,

to

72. A remarkab1e combinatoria1 formula of Heselden , Proc. W.Va. Acad. of Sci. , 73. Eva1uat 土 on of the finite hypergeometric series F(-n ， 1/2;j+1; 的， 7 斗.

The moments of a identities ,

'-~