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8/9/2019 Kajihara Y - Euler Transformation Formula for Multiple Basic Hypergeoemtric Series of Type a and Some Application

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http://www.elsevier.com/locate/aim

Advances in Mathematics 187 (2004) 5397

Euler transformation formula for multiple basic

hypergeometric series of type A and

some applications

Yasushi Kajihara

Department of Mathematics, Kobe University, Rokkodai, Kobe 657-8501, Japan

Received 15 January 2001; accepted 13 August 2003

Communicated by A. Lascoux

Abstract

A multiple generalization of the Euler transformation formula for basic hypergeometric

series 2f1 is derived. It is obtained from the symmetry of the reproducing kernel forMacdonald polynomials by a method of multiple principal specialization. As applications,

elementary proofs of the PfaffSaalschutz summation formula and the Gauss summation

formula for basic hypergeometric series in Un 1 due to S.C. Milne are given. Some othermultiple transformation and summation formulas for very-well-poised 10f9 and 8f7 series,

balanced 4f3 series and 3f2 series are also given.

r 2003 Elsevier Inc. All rights reserved.

MSC: 33D67; 33D52; 33D15; 33C20

Keywords: Multiple basic hypergeometric series of type A; Euler transformation; Macdonald

polynomials; PfaffSaalschutz summation; Watson type transformation; BaileyJackson type transforma-tionsummation formula

1. Introduction

In this paper, we derive a multiple generalization of the Euler transformation

formula for basic hypergeometric series 2f1: We obtain our main result(Theorem 1.1) by specializing suitably the well-known symmetry of the reproducing

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Corresponding author. Present address: Department of Mathematics, Graduate School of Science,

Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 560-0043, Japan. Fax: +81-6-6850-5327.

E-mail address: [email protected].

0001-8708/$ - see front matterr 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.aim.2003.08.012


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kernel for Macdonald polynomials [12]. As applications of our result, we give

elementary proofs of the PfaffSaalschutz summation formula and the Gauss

summation formula for basic hypergeometric series in U

n

1

due to S.C. Milne

[16,17] from the special case of our Euler transformation formula (Proposition 5.1).We also give some other multiple transformation and summation formulas for basic

and ordinary hypergeometric series ( for example, very-well-poised series, Sears

transformation and others).

There are many well-known summation and transformation formulas in the

theory of hypergeometric and basic hypergeometric series. One of the most

important is the binomial theorem

1 uz XkANzkk!

uk; 1:1

where we denote by

zn zz 1?z n 1 1:2

the (ordinary) Pochhammer symbol. A q-analogue of (1.1) is given by

auN

uN

XkAN

akqk

uk: 1:3

Throughout of this paper, we use the notation of q-series [5]

aN

: a; qN

YnAN

1 aqn; ak : a; qk aN

aqkN

for kAC; 1:4

assuming that 0oqo1: We often omit the basis q in q-shifted factorials.On the other hand, one of the most fundamental transformation formulas for

hypergeometric series is the Euler transformation for Gauss hypergeometric series

2F1 (see, for example, [1])

2F1 a; bc

; u 1 ucab2F1 c a; c b

c; u

; 1:5

where

n1Fna0; a1;y; an

b1;y; bn; u

XkAN

a0ka1k?ankk!b1k?bnk

uk: 1:6

Among the most important transformation formulas for basic hypergeometric

series are Heines transformation formulas for 2f1 series.

2f1a; b

c; q; u

bNauNc

NuN

2f1c=b; u

au; q; b

1:7

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Y. Kajihara / Advances in Mathematics 187 (2004) 539754


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c=bNbuNcN

uN

2f1abu=c; b

bu; q; c=b

1:8

abu=cNuN

2f1c=a; c=b

c; q; abu=c

; 1:9

where the basic hypergeometric series r1fr is defined by

r1fra0; a1;y; ar

c1;y; cr; q; u

XnAN

a0na1nyarnqnc1nycrn

un: 1:10

The third equality of Heines transformation formulas (1.9) is a q-analogue of Euler

transformation formula for 2F1 (1.5). It is needless to say that these hypergeometricand basic hypergeometric identities frequently appear in many areas of mathematics

such as in combinatorics, analytic number theory, statistics and representation

theory of Lie algebras.

The multiple basic hypergeometric series in Un 1 initiated by Holman et al.[7,8] have been investigated by many researchers from various viewpoints. The most

important summation formula for basic hypergeometric series in Un 1 is themultiple q-binomial theorem due to S.C. Milne (Theorem 1.47 in [14]):

a1?anuNuN XbANn u

jbj DxqbDx Y1pi; jpn

ajxi=xjbiqxi=xjbi

;

1:11

where jaj Pni1 ai: In this formula,Dx

Y1piojpn

xi xj and Dxqb Y

1piojpn

xiqbi xjqbj

denote the Vandermonde determinants of x x1;y; xn and xqb x1qb1 ;y; xnqbn; respectively. In our previous work [10], we encountered aterminating version of (1.11) when we determine the explicit coefficients of theraising and lowering operators of row type for Macdonald polynomials. We also

gave a proof of (1.11) based on the identity that a constant function 1 is a

eigenfunction of the Macdonalds q-difference operator

Dxu; t; q 1 X

KC1;y;nujKjq

jKj2

YiAK; jeK

1 qxi=xj1 xi=xj

un: 1:12

The terminating version of (1.11) is in fact obtained by a method of multipleprincipal specialization from (1.12). In this paper, we apply the same method to

derive the main result of this paper.

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Y. Kajihara / Advances in Mathematics 187 (2004) 5397 55


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Theorem 1.1. Suppose that none of denominators vanish. Then we have Euler

transformation formula between basic hypergeometric series in Un 1 and Um 1

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn;1pkpm

bkxiyk=xnymgicxiyk=xnymgi

a1?anb1?bmu=cmN

uN

XdANm

a1?anb1?bmu=cmjdj Dyqd

Dy

Y1pk;lpmc=blyk=yldk

qyk=yldk Y1pipn;1pkpmc=aixiyk=xnymdk

cxiyk=xnymdk 1:13

for a11 ;y; a1n ; b1=c;y; bm=cAC:

Remark 1.1. As discussed in [15] more general form, the radius of convergence of the

series in (1.13) is 1.

In this paper, we refer to Eq. (1.13) as the multiple q-Euler transformation. What

is interesting in our multiple q-Euler transformation formula is that the dimension of

the summation in both sides are independent of each other.

In the case when m n 1; formula (1.13) reduces to the third Heinestransformation formula (1.9). We remark that our multiple q-Euler transformation

(1.13) is different from the third Heines transformation for basic hypergeometric

series in Un 1 studied in [6].We give a proof of Theorem 1.1 in Section 3. The source identity for proving our

multiple q-Euler transformation formula for basic hypergeometric series is based on

self-duality of the reproducing kernel for Macdonald polynomials. That is,

Macdonald polynomials and Macdonalds q-difference operators know someproperties of the basic hypergeometric series in Un 1: Our transformationformula of Theorem 1.1 also involves in a sense fundamental properties of

Macdonald polynomials.

We also give some special cases and q-1 limits of (1.13) in Section 4. In

particular, formula (1.13) for m 1 implies an interesting and useful transformationformula (4.1) between n1fn and basic hypergeometric series in Un 1: Eq. (4.1)has also an expression in terms of basic Lauricella function f

nD obtained by

Andrews transformation [2,3].

In fact, formula (1.13) can be considered as a master formula from which various

multiple hypergeometric transformation and summation formulas are derived andwhich provides elementary proofs of them as in usual basic and ordinary

hypergeometric series case.

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In Section 5, we make use of (4.1) to give elementary proofs of the Pfaff

Saalschutz summation formula and the Gauss summation formula for basic

hypergeometric series in U

n

1

by S.C. Milne [16,18].

We also discuss some generalizations of multiple PfaffSaalschutz summationformula in Section 6. In a similar way to our proof of multiple PfaffSaalschutz

summation, we also derive from (1.13) some new multiple transformation and

summation formulas for very-well-poised (basic) hypergeometric series. A general-

ization of a transformation formula between terminating 8W7 series and terminating

balanced 4f3 series and a formula of multiple (basic) hypergeometric series which

involves multiple Jacksons 8W7 terminating summation due to S.C. Milne [17] and

multiple 10W9 transformation formula is included.

In Section 7, we give two type of multiple generalizations of Sears transformation

for balanced 4f3 basic hypergeometric series from the product of two multiple q-

Euler transformations. Two multiple generalizations of terminating 3f2 transforma-

tions are presented.

2. Background information

In this section, we summarize some basic facts on Macdonald polynomials of type

An1 and Andrews transformation formula for basic hypergeometric series betweenf

nD and n1fn: Definitions of very-well-poised (basic) hypergeometric series are also

given.

2.1. Macdonald symmetric polynomials

Here, we define Macdonald polynomials and summarize their properties which we

need in this paper. We follow basically the notation of Macdonalds book [12]. For

detail on Macdonald polynomials, see [12].

Suppose that t is non zero real number and let K Qq; t be the field of rationalfunctions in q; t; and KxSn Kx1;y; xnSn the ring of symmetric functions in nvariables over K

:The Macdonald symmetric polynomials

Plx;

q; tare a family of

homogeneous polynomials that are parametrized by partitions l l1;y; ln withllpn: Macdonalds q-difference operator Dxu; q; t is defined by

Dxu; q; t X

KC1;y;nujKjt

jKj2

YiAK; jeK

1 txi=xj1 xi=xj

YiAK

Tq;xi

Xnr0

urDrx; 2:1

where Tq;xi are the q-shift operators in xi

Tq;xi fx1;y; xi;y; xn fx1;y; xiq;y; xn: 2:2

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This operator is the generating function of the commuting family of q-difference

operators Dr; r 0;y; n: The Macdonald polynomial Plx; q; t are characterized asthe joint eigenfunctions in K

x

Sn of Dr: Namely, each Pl

x; q; t

satisfies the

following q-difference equation

Dxu; q; tPlx; q; t Plx; q; tYni1

1 utniqli: 2:3

Note that Plx; q; t has the leading term mlx under the dominance order ofpartition when it is expressed as a linear combination of monomial symmetric

function mmx: It is known that Plx; q; t is determined uniquely by two conditionsmentioned above.

Macdonald polynomials have a following reproducing kernelYx;y :

Y1pipn;1pkpm

txiykNxiykN

X

llpminn;mblq; tPlx; q; tPly; q; t 2:4

for the variables x x1;y; xn and y y1;y;ym: The coefficient blq; t isdetermined by

blq; t YsAl

1 tls1qas1 tlsqas1: 2:5

The following identity can be proved from (2.3) and (2.4).

Proposition 2.1 (Kirillov and Noumi [11], Mimachi and Noumi [13]). Suppose that

nXm: Then

DxuY

x;y u; tnmDyutnmY

x;y: 2:6

2.2. Andrews transformation formula for basic hypergeometric function

Here, we recall on the multivariable basic hypergeometric function flD and

Andrews transformation formula. Our fundamental reference on the notation of

q-series and basic hypergeometric function is [5].

The Lauricella type multivariable basic hypergeometric series flD is defined as

follows:

flD

a; b1; b2;y; bl

c

; q; x1; x2;y; xl XaANlajajb1a1b2a2yblalcjajqa1qa2yqal

xa;

2:7

where jaj Pli1 ai is the length of multi-index a a1;y; al:

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Andrews derived the transformation formula between flD and l1fl ([2] for l 2;

[3] for any positive integer l).

Proposition 2.2 (Andrews [2,3]).

flD

a; b1; b2;y; bl

c; q; x1; x2;y; xl

aNcN

Ylk1

bkxkNxkN l1

flc=a; x1;y; xl

b1x1;y; blxl; q; a

: 2:8

Note that (2.8) reduces 1st Heines transformation formula when l 1: For ourpurpose, it is convenient to rewrite (2.8) into the following form:

l1fla; b1;y; bl

c1;y; cl; q; u

auNuN

Ylk1

bkNckN

flD

u; c1=b1;y; cl=bl

au; q; b1;y; bl

: 2:9

2.3. Very-well-poised hypergeometric series

The hypergeometric series n1Fn is well-poised if a0 1 a1 b1 ? an bn: It is called very-well-poised if it is well-poised and if a1 a0=2 1: Namely,the very-well-poised n1Fn series is expressed as the following form:

n1Fna0;

a0

2 1; a2; y; an

a0

2; a0 a2 1; y; a0 an 1

; u

264

375

XkANa0 2k

a0

a0ka2k?ankk!a0 a2 1k?a0 an 1k

uk:

2:10

The basic hypergeometric series n1fn is well-poised if

a0q a1c1 ? ancn: It is called very-well-poised if it is well-poised and if a1 q

ffiffiffiffiffia0

pand a2 q ffiffiffiffiffia0p : Namely, the very-well-poised n1fn is expressed as the

following form

n1fna0; q

ffiffiffiffiffia0

p; q ffiffiffiffiffia0p ; a3; y; an

ffiffiffiffiffia0

p;

ffiffiffiffiffia0

p; a0q=a3; y; a0q=an

; q; u

XkAN

1 a0q21 a0

a0ka3k?ankqka0q=a3k?a0q=ank

uk: 2:11

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Hereafter, we employ the notation of very-well-poised (basic) hypergeometric

series as

n1Fna0; a0

2 1; a2; y; an

a0

2; a0 a2 1; y; a0 an 1

; u

264375

n1Vna0; a2;y; an; u 2:12

and

n1fna0; q

ffiffiffiffiffia0

p; q ffiffiffiffiffia0p ; a3; y; an

ffiffiffiffiffia0p

;

ffiffiffiffiffia0

p; a0q=a3; y; a0q=an

; q; u

n1Wna0; a3;y; an; q; u 2:13

for short.

3. Proof of the multiple q-Euler transformation

In this section, we show the main theorem of this paper, Euler transformation

formula between basic hypergeometric series in Un 1 and Um 1 (1.13)XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn;1pkpm


a1?anb1?bmu=cmN

uN

XdANm


Dy

Y1pk;lpm c=bl

yk=yl

dk

qyk=yldkY

1pipn;1pkpm c=ai

xiyk=xnym

dk

cxiyk=xnymdk

by starting from well-known property of Macdonald polynomials in An:First we shall prove

Proposition 3.1. Formula (1.13) is valid for all a1i qai; 1pipn; bk=c qbk;1pkpm with aANn; bANm:

To prove this proposition, we consider the following rational function (see

[11,13]),

Fujz; w Y

z; w1DzuY

z; w 3:1

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for the variables z z1;y; zr and w w1;y; wp: The rational function Fujz; wis of the form

Fujz; w X

KC1;y;rujKjt jKj2

YiAK; jeK

1 tzi=zj1 zi=zj

YiAK;1pkpp

1 ziwk1 tziwk: 3:2

We now assume rXp: Note that the rational function Fujz; w has a followingself-duality from (2.6)

Fujz; w u; trpFutrpjw; z: 3:3For convenience of latter discussion, replace t to q: For each multi-index aANn

with jaj r; we define the multiple principal specialization y y1;y;yr to thepoints pav; x by

y y1;y;yr

-pav; x v=x1;v=x1q;y;v=x1qa11;v=x2;y;

v=xn;y;v=xnqan1: 3:4We analyze some properties for multiple principal specialization. When we specialize

z to pav; x; the index set 1;y; r is divided into n blocks with cardinality a1;y; an;respectively. Note also that

YiAK; jeK

qzi zjzi zj

zpav;x

is equal to zero unless the elements of K should be packed to the left in each block.

Simple verification leads to the following lemma.

Lemma 3.1. For such a configuration K; we denote the number of points of K sitting ini-th block for i 1;y; n as gi and set g g1;y; gn: We have

YiAK; jeK

qzi

zj

zi zj zpav;x YiAK; jeKqzi

zj

zi zj zpa1;x qjajjgjq

jgj2

DxqgDx

Y1pi; jpn

qajxi=xjgiqxi=xjgi

3:5

for any v and

YiAK

Ymk1

1 ziwk1 qziwk

zpa1;x;wpbq

1;y qjbjjgj q

bjxiyjgixiyjgi

: 3:6

Proof of Proposition 3.1. We first replace t in (3.3) by q; and then specialize z and w

to pa1; x and pbq1;y; respectively, where x x1;y; xn; y y1;y;ym and

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aANn; bANm are multi-indices with jaj r and jbj p; respectively. Then we have

XgANn;gpa

qjajjbjujgj Dxqg

Dx

Y

1pi; jpn

qajxi=xjgiqxi=xjgi

Y1pipn;1pkpm

qbkxiykgixiykgi

ujajjbjX

dANm;dpb

ujdj Dyqd

Dy

Y1pk;lpm

qblyk=yldkqyk=yldk Y1pipn;1pkpm

qaixiykdkxiykdk

: 3:7

Then by changing the variables and parameters u-uqjbjjaj; xi-xi=xn 1pipn;yk-cyk=ym 1pkpm to (3.7), we obtain the desired identity. Thus we complete theproof of the proposition.

Proof of the main theorem. To prove the main theorem, by replacing u-uqjbj; werewrite (3.7) as follows:

uqjajN XgANn

ujgjqjgj2

Y1pi; jpn

qaigj1xi=xjgj

qgigj1xi=xj

gj Y1pipn;1pkpm

qbkxiykgi

xiyk

gi

uqjbjN

XdANm

ujdjqjdj2

Y1pk;lpm

qbkdl1yk=yldlqdkdl1yk=yldl

Y

1pipn;1pkpm

qaixiykdkxiykdk

: 3:8

By taking the coefficient of us in (3.8), we obtain

XgANn;jgjps

1

jgj

qjgj2 Y

1pi; jpn

qaigj1xi=xjgjqgigj1xi=xjgj Y1pipn;1pkpm

qbkxiykgixiykgi

qjajsjgjqsjgj

2

X

dANm;jdjps1jdjq

jdj2

Y1pk;lpm

qbkdl1yk=yldlqdkdl1yk=yldl

Y1pipn;1pkpm

qaixiykdkxiykdk

qjbjsjdjqsjdj

2

: 3:9

Note that this is a polynomial identity of qai; 1pipn and qbk; 1pkpm: Hence (1.13)is valid for all a1i 1pipn; bk=c 1pkpmAC for formal power series of u: Thusthe main theorem is proved.

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Remark 3.1. The homogeneous version of the multiple q-Euler transformation (1.13)

is expressed as

ut1?tnNXgANn

ut1?tnjgj DxqgDx

Y

1pi; jpn

t1j xi=xjgiqxi=xjgi

Y1pipn;1pkpm

skxiykgixiykgi

us1?smNXdANm

us1?smjdj Dyqd

Dy

Y1pk;lpms1l yk=yldkqyk=yldk Y1pipn;1pkpm

tixiykdkxiykdk

:

3:10

4. Some special and limiting cases of multiple q-Euler transformation

In this section, we give some formulas of special cases of our Euler transformation

formula (1.13)

XgANn

ujg

jD

xqg

Dx Y1pi; jpn

ajxi=xj

gi

qxi=xjgiY

1pipn;1pkpm

bkxiyk=xnym

gi

cxiyk=xnymgi a1?anb1?bmu=c

mN

uN

XdANm


Dy

Y

1pk;lpm

c=blyk=yldkqyk=yldk

Y1pipn;1pkpm

c=aixiyk=xnymdkcxiyk=xnymdk

and the corresponding q-1 limits.

Proposition 4.1. In the case when m 1; the multiple q-Euler transformation formula(1.13) reduces to

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

a1?anbu=cN

uN

n1fnc=b; c=a1x1=xn;y; c=an1xn1=xn; c=an

cx1=xn;y; cxn1=xn; c; q; a1yanbu=c

: 4:1

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Furthermore, This formula has an alternative expression in terms of Lauricella basic

hypergeometric function

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

a1?anuNuN

Y1pipn

c=aixi=xnNcxi=xnN

fnDa1?anbu=c; a1;y; an

a1?anu; q; c=a1x1=xn;y; c=an1xn1=xn; c=an

: 4:2

Formula (4.2) is obtained from (4.1) by using Andrews transformation (2.9).

Remark 4.1. In the case when n 1; (4.2) reduces to the 2nd Heines transformationformula (1.8).

Formula (4.1) will be a key to giving elementary proofs of some summation

formulas for basic hypergeometric series in Un 1 in the next section of thepresent paper.

Remark 4.2. In the case when m n and yk x1

k ; (1.13) reduces to

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

bjxi=xjgicxi=xjgi

a1?anb1?bnu=cnN

uN

XdANn

a1?anb1?bnu=cnjdj Dx1qd

Dx1

Y1pi; jpn

c=bixj=xidiqxj=xidic=aixi=xjdicxi=xjdi

: 4:3

The informed reader might compare this formula with the third Heines

transformation formula for basic hypergeometric series in Un 1 due toGustafson and Krattenthaler [6].

Now, we will write down the q-1 limit of the formulas in this section.

To this end, note that

limq-1

1 qz1 q z: 4:4

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As an application of (4.4), we have

limq-1

qzn1 qn

zn:

4:5

Note also that

limq-1

qzuN

uN

1 uz 4:6

and

limq-1

qN

qsN

1 qs1 Gs; 4:7

where Gs is the Euler gamma function. Before taking the limit, we replace all theparameters z in our formulas by qz:

Then, by taking the limit q-1 in Eq. (1.13), we have the Euler transformation for

multiple hypergeometric series of type A:

Proposition 4.2.

XgANn

ujgjDx gDx

Y1pi; jpn

aj xi xjgi1 xi xjgi

Y1pipn;1pkpm

bk xi yk xn ymgic xi yk xn ymgi 1 umc

P1pipn

aiP

1pkpmbk

XdANm

ujdjDy dDy

Y1pk;lpm

c bl yk yldk1 yk yldk

Y1pipn;1pkpm

c ai xi yk xn ymdkc xi yk xn ymdk

: 4:8

Corollary 4.1. In the case when m 1; our Euler transformation formula for multiple(ordinary) hypergeometric series (4.8) reduces to

XgANn

ujgjDx gDx

Y1pi; jpn

aj xi xjgi1 xi xjgi

Y1pipn

b xi xngic xi xngi

1 ucbP

1pipnai

n1Fn c b; c a1 x1 xn;y; c an1 xn1 xn; c an

c x1 xn;y; c xn1 xn; c; u

: 4:9

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Remark 4.3. The q-1 limit of (4.3) is

XgANn uj

g

jD

x

g

Dx Y1pi; jpnaj xi xjgi1 xi xjgi

bj xi xjgic xi xjgi

1 uncP

1pipnaibi X

dANn

ujdjDx dDx

Y

1pi; jpn

c bi xj xidi1 xj xidi

c ai xi xjdic xi xjdi

: 4:10

Remark 4.4. The homogeneous version of multiple Euler transformation (4.8) is

expressed as

1 uP

itiXgANn

ujgjDx gDx

Y

1pi; jpn

ti xi xjgi1 xi xjgi

Y1pipn;1pkpm

sk xi ykgixi ykgi

1 uP

ksk

XdANm

ujdjDy dDy

Y1pk;lpm

sl

yk

yl

dk

1 yk yldkY

1pipn;1pkpmti

xi

yk

dk

xi ykdk : 4:11

5. Simple proofs of some summation formulas for basic hypergeometric series in

Un 1

In this section, we give simple proofs of the PfaffSaalschutz summation formula

and the Gauss summation formula for basic hypergeometric series in Un 1 byusing the special case of our Euler transformation formula for multiple basichypergeometric series (4.1)

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

a1?anbu=cNuN


cx1=xn;y; cxn

1=xn; c; q; a1y; anbu=c :

By virtue of this formula, our proofs are essentially the same as in the case of basic

hypergeometric series of one variable.

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Proposition 5.1 (PfaffSaalschutz summation formula for basic hypergeometric

series in Un 1).X

jgjANnqjgj Dxqg

DxY

1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

qNjgja1?anbq1N=cjgj

c=bNc=a1?anbNY

1pipn

c=aixi=xnNcxi=xnN

: 5:1

Remark 5.1. This formula is already known by S.C. Milne ([18, Theorem 4.15]). In

the case when n 1; (5.1) reduces to the ordinary PfaffSaalschutz summationformula for terminating balanced 3f2 series ( formula (1.7.2) in [5])

3f2a; b; qN

c; q1Nab=c; q; q

c=aNc=bNcNc=abN

: 5:2

Proof. We rewrite Eq. (4.1) as

cu=a1?anbN

u

N XgANn

cu=a1?anbjgj Dxqg

D

x

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi


cx1=xn;y; cxn1=xn; c; q; u

: 5:3

By taking the coefficient of uN of the equation above, we have

XgANn;jgjpl

c=a1?anb

jgj Dxqg

Dx Y1pi; jpnajxi=xjgiqxi=xjgi

Y

1pipn

bxi=xngicxi=xngi

c=a1?anbNjgjqNjgj

c=bNqNY

1pipn

c=aixi=xnNcxi=xnN

: 5:4

Note that

z

N

r

qNr z

N

qN qN

r

z1q1Nrq

z r

: 5:5Then a simple verification leads to the desired identity (5.1). &

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By taking the limit l-N; we have

Proposition 5.2 (Gauss summation formula for basic hypergeometric series in

Un 1).

XgANn

c=a1?anbjgj Dxqg

DxY

1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

c=bNc=a1?anbNY

1pipn

c=aixi=xnNcxi=xnN

: 5:6

Remark 5.2. This formula is already known by S.C. Milne ([16, Theorem 3.9]). Inthe case when n 1; Eq. (5.6) reduces to the ordinary Gauss summation formula for2f1 series ( formula (1.5.1) in [5])

2f1a; b;

c;; q; c=ab

c=aNc=bNc

Nc=ab

N

: 5:7

Similarly, from (4.9)

XgANn

ujgjDx gDx

Y1pi; jpn

aj xi xjgi1 xi xjgi

Y1pipn

b xi xngic xi xngi

1 ucb

P1pipn

ai

n1Fnc b; c a1 x1 xn;y; c an1 xn1 xn; c an

c

x1

xn;y; c

xn

1

xn; c

; u ;

the PfaffSaalschutz summation formula

Xjgjpl

DxgDx

Y1pi; jpn

aj xi xjgi1 xi xjgi

Y1pipn

bxi xngic xi xngi

Njgj

1

N

a1

?

an

b

c

jg

j c bNc a1 ? an bN

Y1pipn

c ai xi xnNc xi xnN

5:8

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and the Gauss summation formula

XgANn

D

x

g

Dx Y1pi; jpnaj xi xjgi1 xi xjgi Y1pipn

b xi xngic xi xngi

Gc a1 ? an bGc b

Y1pipn

Gc xi xnGc ai xi xn 5:9

for hypergeometric series in Un 1 are derived directly in the same way as in thecase of basic hypergeometric series in Un 1 case.

Remark 5.3. Eqs. (5.8) and (5.9) are the q-1 limiting cases of (5.1) and (5.6),

respectively. In the case when n 1; (5.8) reduces to the PfaffSaalschutz summationformula for balanced terminating

3F

2series

3F2N; a; b

c; a b c N 1 ; 1

c aNc bNcNc a bN5:10

and (5.6) reduces to the Gauss summation formula for Gauss hypergeometric

series 2F1

2F1a; b

c; 1

GcGc a bGc aGc b: 5:11

6. Beyond the q-PfaffSaalschutz summation formula in Un 1

In this section, we discuss some generalizations of multiple q-PfaffSaalschutz

formula by using the multiple q-Euler transformation formula (1.13). As a result, we

derive some multiple transformation formulas for very-well-poised basic hypergeo-

metric series and recover a multiple generalization of the Jacksons 8W7 summation

theorem due to Milne [17] by using the same method as in the previous section.

Before preceding our derivations, let us explain the essence of our discussion. Note

that, the homogeneous part of the summation (the summand is taken by all the

multiindices bANn1 with jbj N for any nonnegative integer N) in the multiplebasic hypergeometric series is of the form

XbANn1;jbjN

DxqbDx u

b11 ?u

bn1n1

basic hypergeometric stuff

XbANn

D

xqb

Dx Y1pipn

1

aqjbjbixi=xn

1 axi=xn zb11 ?z

bnn

another basic hypergeometric stuff; 6:1

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where we notice that bn1 N b1 ? bn: Furthermore, the case when n 1;that series in right-hand side of equation above is reduced to very-well-poised basic

hypergeometric series which are introduced in Section 2. In the course of his

derivation of multiple q-binomial theorem, Milne [14] proved a multiple general-ization of Rogers terminating very-well-poised 6W5 summation theorem. Con-

versely, after changing n-n 1; take the coefficients ofuN in the both side of (1.11)b1?bnbn1uN

uN

X

bANn1ujbj Dxq

bDx

Y1pi; jpn1

bjxi=xjbiqxi=xjbi

:

Then by changing the parameter suitably, we obtain multiple Rogers terminating

very-well-poised 6W5 summation formula [18],

aq=b1?bncNaq=cN

Y1pipn

aqxi=xnNaq=bixi=xnN

XbANn

aq1N

b1?bnc

jbjDxqbDx

Y1pipn

1 aqjbjbixi=xn1 axi=xn

Y

1pi; jpn

bjxi=xjbiqxi=xjbi

Y1pipn

cxi=xnbiaq1Nxi=xnbi

qNjbjaq=cjbj Y1pipn

axi=xnjbjaq=bixi=xnjbj

:

6:2

Note that in the case when n 1; (6.2) reduces to the Rogers terminating very-well-poised 6W5 summation (cf. [5])

aq=bcNaq=cN

aqNaq=bN

6W5 a; b; c; qN; q; aq1N

bc

: 6:3

Multiple q-binomial theorem and our Euler transformation formula (1.13) can be

interpreted as a generating functions of (multiple) very-well-poised hypergeometric

series. We shall carry out the latter one in several cases.

6.1. Multiple generalization of Watson type transformation formula

In this subsection, we give a multiple transformation formula of Watson type for

very-well-poised basic hypergeometric series (see Proposition 6.1 below). As a special

case, it implies the following transformation formula between a 4f3 series and a very-

well-poised 8W7 series:

8W7 a; b; c; d; e; qN; q;

a2qN2

bcde a

2q2=bcdeNeNaqNaq=bNaq=cNaq=dN4

f3qN; aq=be; aq=ce; aq=deq1N=e; a2q2=bcde; aq=e

; q; q

: 6:4

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Formula (6.4) is similar to a Watson transformation formula (Eq. (2.5.1) in [5])

8W7 a; b; c; d; e; qN

; q;

a2q2N

bcde

aqNaq=deNaq=dNaq=eN4f3

qN; d; e; aq=bcaq=b; aq=c; deqN=a

; q; q

: 6:5

In fact, (6.4) includes the same number of free parameters as (6.5) and right-hand

side of (6.4) and (6.5) are balanced. Formula (6.5) may be obtained by combining

Sears transformation (see next section) from (6.4). In this paper, we call (6.4) a

Watson type transformation formula. The informed reader might compare some

formula for multiple Watson transformation formula in Milne [17], MilneLily [19]

and Schlosser [22].

Proposition 6.1

an1qn1=b1?bmcd1?dnenNaq=cN

Y1pkpmaqyk=ymN

aq=bkyk=ymN Y1pipnexn=xiN

aq=dixn=xiN

XgANn;jgjpN

qjgjDxqgDx

Y1pi; jpn

aq=djexi=xjgiqxi=xjgi

Y

1pipn;1pkpm

aq=bkexiyk=xnymgiaq=exiyk=xnymgi

Y1pipn

aq=cexi=xngiq1Ne1xi=xngi

qNjgj

an1qn1=b1?bmcd1?dnenjgj X

dANma

n

1

qN

n

1

=b1?bmcd1?dnen

jdj

Dyqd

DyY

1pkpm

1 aqjdjdkyk=ym1 ayk=ym

Y

1pk;lpm

blyk=yldkqyk=yldk

Y1pkpm

cyk=ymdkaqNyk=ymdk

Y1pipn;1pkpm

dixiyk=xnymdkaq=exiyk=xnymdk

qNjdj

aq=cjdjY

1pkpm

ayk=ymjdjaq=bkyk=ymjdj

Y1pipn

exn=xijdjaq=dixn=xijdj

: 6:6

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Proof. By taking the coefficient of uN in

cm1u=a1?anb1?bmbm1NuN

XgANn

cm1u=a1?anb1?bmbm1jgj Dxqg

Dx

Y

1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn;1pkpm1

bkxiyk=xnym1gicxiyk=xnym1gi

XdANm1

ujdjD yqdD

y Y1pk;lpm1

c=blyk=yldk

qyk=yldk

Y

1pipn;1pkpm1

c=aixiyk=xnym1dkcxiyk=xnym1dk

; 6:7

where y y1;y;ym;ym1; we have

cm1=a1?anb1?bmbm1Nc=bm1N Y1pkpm

ym1=ykNc=bkym1=ykN

Y

1pipn

cxi=xnNc=aixi=xnN

X

gANn;jgjpNqjgj

DxqgDx

Y1pi;jpn

ajxi=xjgiqxi=xjgi

Y

1pipn;1pkpm


Y1pipn

bm1xi=xngicxi=xngi

qN

jgjq1Na1?anb1?bmbm1=cm1jgj

XdANm

a1?anb1?bmbm1q=cm1jdj

Dyqd

DyY

1pkpm

1 qNjdjdkyk=ym11 qNyk=ym1

Y1pk;lpm

c=blyk=yldkqyk=yldk Y1pkpm

c=bm1yk=ym1dkqyk=ym1dk

Y

1pipn;1pkpm

c=aixiyk=xnym1dkcxiyk=xnym1dk

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qNjdj

q1Nbm1=cjdjY

1pkpm

qNyk=ym1jdjq1Nbk=cyk=ym1jdj

Y

1pipn

q1Nc1xn=xijdjq1Nai=cxn=xijdj

: 6:8

After some simple verification in the equation above, we obtain

cm1=a1?anb1?bmbm1Nc=bm1N

Y1pkpm

ym1=ykNc=bkym1=ykN

Y1pipn

cxi=xnNc=aixi=xnN

X

gANn;jgjpNqjgj

DxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y

1pipn;1pkpm


Y1pipn

bm1xi=xngicxi=xngi

qNjgj

q1

Na

1?a

nb

1?b

mb

m1=cm

1

jgjXdANm

a1?anb1?bmbm1q=cm1jdj

Dyqd

DyY

1pkpm


Y

1pk;lpm

c=blyk=yldkqyk=yldk

Y1pkpm


Y1pipn;1pkpmc=aixiyk=xnym1dk

cxiyk=xnym1dk

qNjdj

q1Nbm1=cjdjY

1pkpm


Y

1pipn


: 6:9

Finally by changing the parameters suitably, we have the desired formula. &

In the case when m 1; the right-hand side of Eq. (6.6) reduces to the specialvalue of very-well-poised basic hypergeometric series.

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Corollary 6.1.

an1qn1=bcd1?dnen

N

aq=cN aq

N

aq=bN Y1pipn exn=xi

N

aq=dixn=xiN

X

gANn;jgjpNqjgj

DxqgDx

Y1pi; jpn

aq=djexi=xjgiqxi=xjgi

Y

1pipn

aq=bexi=xngiaq=cexi=xngiaq=exi=xngiq1Ne1xi=xngi

" #

qNjgj

an1qn1=bcd1?dnenjgj 2n6W2n5a; b; c; d1x1=xn;?dn1xn1=xn; dn;

exn=x1;?exn=xn1; e; qN; q; an1qNn1=bcd1?dnen: 6:10

In the case when n 1; Eq. (6.6) reduces to

Corollary 6.2.

a2q2=b1?bmcdeNaq=cN

exNaq=dN

Y1pkpm

aqyk=ymNaq=bkyk=ymN

m3fm2qN; aq=b1ey1=ym;y;

a2q2=b1?bmcde; aq=ey1=ym;y;

aq=bm1eym1=ym; aq=bme; aq=ce; aq=de

aq=eym1=ym; aq=e; q1N=e; q; q

XdANm

a2qN2=b1?bmcdejdj

DyqdDy

Y1pkpm

1 aqjdjdkyk=ym1 ayk=ym

Y

1pk;lpm

blyk=yldkqyk=yldk

Y1pkpm

cyk=ymdkdyk=ymdkaqNyk=ymdkaq=eyk=ymdk

" #

qNjdjejdj

aq=cjdjaq=djdjY

1pkpm

ayk=ymjdjaq=bkyk=ymjdj

: 6:11

Remark 6.1. Eq. (6.6) is a multiple generalization of Watson type transformation.

Namely, (6.4) is a n 1; m 1 case of (6.6).

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6.2. BaileyJackson type transformation-summation formula

We give a multiple transformation-summation formula of very-well-poised basic

hypergeometric series including, as special cases, Jacksons terminating 8W7summation formula

8W7a; b; c; d; e; qN; q; q

aqNaq=bcNaq=bdNaq=cdNkq=bNaq=cNaq=dNaq=bcdN6:12

when a2qN1 bcde (Eq. (2.6.1) in [5]), and 10W9 transformation formula

aq=bNaq=cNaq=dNaq=eNmqNmf=aNmbf=aNmcf=aNmdf=aNmef=aNaqN fN 10W9a; b; c; d; e;f; mfqN; qN; q; q

10W9m; aq=bf; aq=cf; aq=df; aq=ef; mf=a; mfqN; qN; q; q; 6:13

where m a3q2=bcdef2:Note that (6.13) is similar to the Bailey transformation formula for a terminating

balanced 10W9 series (Eq. (2.9.1) in [5])

10W9a; b; c; d; e;f; laqN1=ef; qN; q; q

aq

N

aq=ef

N

lq=e

N

lq=f

N

aq=eNaq=fNlqNlq=efN 10W9l; lb=a; lc=a; ld=a; e;f; laqN1=ef; qN; q; q; 6:14

where l a2q=bcd:Though it has same conditions (number of free parameters and series of both side

are balanced) as in Baileys 10W9 transformation (6.14), formula (6.13) is different

from known ones. One can check that both the Bailey transformation (6.14) and our

10W9 transformation (6.13) can be obtained by iterating twice the another one. For

comparison to other multiple 10W9 transformation formula in An; see MilneNewcomb [20] and DenisGustafson [4].

In this paper, we call (6.13) the Bailey type transformation.

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Proposition 6.2.

XgANn;jgjN

DxqgDx Y1pi; jpn

ajxi=xjgiqxi=xjgi Y1pipn;1pkpm


X

dANm;jdjN

DyqdDy

Y1pk;lpm

c=blyk=yldkqyk=yldk

Y

1pipn;1pkpm


6:15

when a1?anb1?bm cm:

Proof. In the case when a1?anb1?bm cm; (1.13) is written asXgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn;1pkpm


XdANm

ujdjDyqdDy

Y1pk;lpm

c=blyk=yldkqyk=yldk

Y1pipn;1pkpm


: 6:16

By taking the coefficient of uN in (6.16), we have the desired formula. &

In this paper, we call (6.15) the BaileyJackson type transformation-summation

formula.

Now we give a multiple generalization of Bailey type transformation (6.13).

Corollary 6.3.

aq=d

N

aq=e

N

mdf=aNmef=aNY

1pkpmmqyk=ym

N

aq=ck

ym=yk

N

mckf=ayk=ymN fym=ykN

Y1pipn

aq=bixi=xnNmf=axn=xiNaqxi=xnNmbif=axn=xiN

XgANn

qjgjDxqgDx

Y1pipn

1 aqjgjgixi=xn1 axi=xn

Y1pi; jpn

bjxi=xjgiqxi=xjgi Y1pipn

dxi=xngiaqN1xi=xngi

Y

1pipn;1pkpm

ckxiyk=xnymgiaq=fxiyk=xnymgi

Y1pipn

exi=xngiaq1N=mfxi=xngi

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qNjgj

aq=djgjY

1pipn

axi=xnjgjaq=bixi=xnjgj

mfqNjgjaq=ejgjY

1pkpm

fym=ykjgjaq=ckym=ykjgj

XdANm

qjdjDyqdDy

Y1pkpm1

1 mqjdjdkyk=ym1 myk=ym

Y

1pk;lpm

aq=clfyk=yldkqyk=yldk

Y1pkpm

aq=efyk=ymdkmqN1yk=ymdk

Y1pipn;1pkpmaq=bi fxiyk=xnymdkaq=fxiyk=xnymdk Y1pkpm

aq=dfyk=ymdkq

1

N

=fyk=ymdk

qNjdj

mef=ajdjY

1pkpm

myk=ymjdjmck f=ayk=ymjdj

mfqNjdj

mdf=ajdjY

1pipn

mf=axn=xijdjmbi f=axn=xijdj

; 6:17

where m am2qm1=b1?bnc1?cmdefm1:

Proof. Replace n to n 1 and m to m 1 in (6.15). Some simple but longmanipulation leads to the following in the case when a1?anan1b1?bmbm1 cm1

an1Nc=an1N

bm1Nc=bm1N

Y1pkpm

ym1=ykNc=bkym1=ykN

bkyk=ym1Ncyk=ym1N

Y

1pipn

cxi=xn1Nc=aixi=xn1N

aixn1=xiNxn1=xiN

XgANn qjg

jD

xqg

Dx Y1pipn1

qNjgjgixi=xn1

1 qNxi=xn1

Y

1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

an1xi=xn1giqxi=xn1gi

Y

1pipn;1pkpm

bkxiyk=xn1ym1gicxiyk=xn1ym1gi

Y1pipn

bm1xi=xn1gicxi=xn1gi

qNjgj

q1Na1n1jgj Y1pipnqNxi=xn1jgj

q1Na1i xi=xn1jgj

Y

1pkpm

q1Nc1ym1=ykjgjq1Nb1k ym1=ykjgj

q1Nc1jgjq1Nb1m1jgj

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X

dANm;jdjpNqjdj

DyqdDy

Y1pkpm


Y

1pk;lpm

c=blyk=yldkqyk=yldk

Y1pkpm1


Y

1pipn;1pkpm

c=aixiyk=xn1ym1dkcxiyk=xn1ym1dk

Y1pkpm

c=an1yk=ym1dkcyk=ym1dk

qNjdj

q1Nbm1=cjdjY

1pkpm


Y1pipnq1Nc1xn1=xijdj

q1

N

ai=cxn1=xijdjq1Nc1jdj

q1

N

an1=cjdj:

6:18

Finally by changing the parameters suitably, we have the desired identity (6.17). &

Corollary 6.4. In the case when m 1; (6.17) reduces to

mqNaq=cNaq=dNaq=eNfNmcf=aNmdf=aNmef=aN Y

1pipn

aq=bixi=xnNmf=axn=xiNaqxi=xnNmbi f=axn=xiN

XgANn

qjgjDxqgDx

Y1pipn

1 aqjgjgixi=xn1 axi=xn

Y1pi;jpn

bjxi=xjgiqxi=xjgi

Y

1pipn

cxi=xngidxi=xngiexi=xngiaq=fxi=xngiaqN1xi=xngiaq1N=mfxi=xngi

" #

fjgjmfqNjgjqNjgjaq=cjgjaq=djgjaq=ejgj Y1pipn


2n8W2n7m; aq=b1 fx1=xn;y; aq=bn1 fxn1=xn; aq=bn f; aq=cf;

aq=df; aq=ef; mf=axn=x1;y; mf=axn=xn1; mf=a; mfqN; qN; q; q 6:19

where m a3q2=b1?bncdef2:

Remark 6.2. m n 1 case of (6.17) is (6.13).

In the case when m 1; (6.15) reduces to the following multiple generalization ofJackson summation (6.12) by some manipulations and change of parameters as in

multiple Bailey type transformation.

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Corollary 6.5.

XgANn

qjg

jD

xqg

Dx Y1pipn

1

aqjgjgixi=xn

1 axi=xn

Y

1pi; jpn

bjxi=xjgiqxi=xjgi

Y1pipn

cxi=xngiexi=xngiaqN1xi=xngiaq=dxi=xngi

" #

qNjgj

aq=cjgjdjgj

aq=ejgjY

1pipn


aq=b1?bncNaq=cdNaq=b1?bncdNaq=cN Y1pipnaq=bidxi=xnNaqxi=xnN

aq=bixi=xnNaq=dxi=xnN6:20

provided a2qN1 b1?bncde:

Remark 6.3. Formula (6.20) is already known by Milne (Theorem 4.70 in [17]). The

n 1 case of (6.20), namely m 1; n 2 of (6.15) is (6.12).

6.3. q-1 limits

We can also obtain the results for ordinary hypergeometric series from (4.8) in thesame way as in basic hypergeometric case. Here, we state the q-1 limits of the

transformation and summation formulas of this section.

First, we give some q-1 limits of Watson type transformations.

Proposition 6.3. The q-1 limit of (6.6) is

n 1a n 1 b1 ? bm c d1 ? dn neNa 1 cN

Y1pkpm

a

1

yk

ym

N

a 1 bk yk ymNY

1pipne

xn

xi

N

a 1 di xn xiN

X

gANn;jgjpN

Dx gDx

Y1pi; jpn

a 1 dj e xi xjgi1 xi xjgi

Y

1pipn;1pkpm

a 1 bk e xi yk xn ymgia 1 e xi yk xn ymgi

Y1pipn

a 1 c e xi yk xn ymgi

1

N

e

xi

yk

xn

ym

gi

Njgjn 1a n 1 b1 ? bm c d1 ? dn nejgj

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XdANm

Dy dDy

Y1pkpm

a jdj dk yk yma yk ym

Y1pk;lpm

bl yk yldk1 yk yldkY

1pkpm

c yk ymdka N yk ymdk

Y

1pipn;1pkpm

di xi yk xn ymdka 1 e xi yk xn ymdk

Njdja 1 cjdjY

1pkpm

a yk ymjdja 1 bk yk ymjdj

Y1pipn

e xn xijdj

a

1

di

xn

xi

jdj: 6:21

In the case when m 1; the right-hand side of Eq. (6.21) reduces to the specialvalue of very-well-poised hypergeometric series.

Corollary 6.6

n 1a n 1 b c d1 ? dn neN

a

1

c

N

a 1Na 1 bNY

1pipn

e xn xiNa 1 di xn xiN

X

gANn;jgjpN

Dx gDx

Y1pi; jpn

a 1 dj e xi xjgi1 xi xjgi

Y

1pipn

a 1 b e xi xngia 1 c e xi xngia 1 e xi xngi1 N e xi xngi

" #

N

jg

jn 1a n 1 b c d1 ? dn nejgj 2n5V2n4a; b; c; d1 x1 xn;?dn1 xn1 xn; dn;

e xn x1;?e xn xn1; e;N; 1: 6:22

Corollary 6.7. In the case when n 1; Eq. (6.21) reduces to

2a 2 b1 ? bm c d eN

a

1

c

N

eNa 1 dNY

1pkpm

a 1 yk ymNa 1 bk yk ymN

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m3Fm2N; a 1 b1 e y1 ym;y;

2a 2 b1 ? bm c d e; a 1 e y1 ym;y;

a 1

bm1 e ym1 ym; a 1

bm e; a 1

c e; a 1

d ea 1 e ym1 ym; a 1 e; 1 N e ; 1

XdANm

Dy dDy

Y1pkpm

a jdj dk yk yma yk ym

Y1pk;lpm

bl yk yldk1 yk yldk

Y

1pkpm

c yk ymdkd yk ymdka N yk ymdka 1 e yk ymdk

" #

Njdjejdj

a

1

c

jd

ja

1

d

jd

j Y1pkpma yk ymjdj

a

1

bk

yk

ym

jd

j

: 6:23

Remark 6.4. In the case when m n 1 of (6.21) is

7V6a; b; c; d; e;N; 1

2a 2 b c d eNeNa 1Na 1 bNa 1 cNa 1 dN 4F3

N; a 1 b e; a 1 c e; a 1 d e1

N

e; 2a

2

b

c

d

e; a

1

e

; 1 6:24This formula is similar to but different from the Whipple transformation formula

between terminating very-well-poised 7F6 series and terminating 4F3 series

(Eq. (2.4.1.1) in [23])

a 1Na 1 d eNa 1 dNa 1 eN 4

F3N; d; e; a 1 b c

a 1 b; a 1 c; d e a N; 1

7V6a; b; c; d; e;N; 1: 6:25

Next, we present some q-1 limit of multiple Bailey-Jackson type transformation-summation formula and some special cases.

The q-1 limit of multiple Bailey-Jackson type transformation-summation

formula is the following.

Proposition 6.4. When a1 ? an b1 ?bm mc; we have

XgANn;jgjNDx gDx Y1pi; jpn

aj xi xjgi1 xi xjgi

Y

1pipn;1pkpm

bk xi yk xn ymgic xi yk xn ymgi

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X

dANm;jdjN

Dy dDy

Y1pk;lpm


Y

1pipn;1pkpm

c ai xi yk xn ymdkc xi yk xn ymdk

: 6:26

Now we give a q-1 limit of the multiple generalization of Bailey type

transformation (6.17).

Corollary 6.8.

a 1 dNa 1 eNm d f aNm e f aN

Y1pkpm

m 1 yk ymNa 1 ck ym ykNm ck f a yk ymN f ym ykN

Y

1pipn

a 1 bi xi xnNm f a xn xiNa 1 xi xnNm bi f a xn xiN

XgANn

Dx gD

x

Y1pipna jgj gi xi xn

a

xi

xn

Y

1pi; jpn

bj xi xjgi1 xi xjgi

Y1pipn

d xi xngia N 1 xi xngi

Y

1pipn;1pkpm

ck xi yk xn ymgia 1 f xi yk xn ymgi

Y

1pipn

e xi xngia 1 N m f xi xngi

N

jgja 1 djgj

Y1pipn

a

xi

xn

jgja 1 bi xi xnjgj

m f Njgja 1 ejgjY

1pkpm

f ym ykjgja 1 ck ym ykjgj

XdANm

Dy dDy

Y1pkpm

m jdj dk yk ymm yk ym

Y1pk;lpm

a 1 cl f yk yldk1 yk yldk Y1pkpm

a 1 e f yk ymdkm N 1 yk ymdk

Y

1pipn;1pkpm

a 1 bi f xi yk xn ymdka 1 f xi yk xn ymdk

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Y

1pkpm

a 1 d f yk ymdk1 N f yk ymdk

Njdjm e f ajdjY

1pkpm

m yk ymjdjm ck f a yk ymjdj

m f Njdjm d f ajdjY

1pipn

m f a xn xijdjm bi f a xn xijdj

;6:27

where m m 2a m 1 b1 ? bn c1 ? cm d e m 1f:

Corollary 6.9. In the case when m 1; (6.27) reduces tom 1Na 1 cNa 1 dNa 1 eN

fNm c f aNm d f aNm e f aN

Y

1pipn

a 1 bi xi xnNm f a xn xiNa 1 xi xnNm bi f a xn xiN

XgAN

n

Dx gD

x Y1pipn

a jgj gi xi xna

x

i x

nY

1pi; jpn

bj xi xjgi1

x

i x

jgi

Y

1pipn

c xi xngid xi xngie xi xngia 1 f xi xngia 1 N m f xi xngia N 1 xi xngi

" #

fjgjm f NjgjNjgja 1 cjgja 1 djgja 1 ejgjY

1pipn

a xi xnjgja 1 bi xi xnjgj

2n5V2n4m; a 1 b1 f x1 xn;y; a 1 bn1 f xn1 xn;

a 1 bn f; a 1 c f; a 1 d f; a 1 e f;

m f a xn x1;y; m f a xn xn1; m f a; m f N;N; 1

6:28where m 3a 2 b1 ? bn c d e 2f:

Remark 6.5. The m n 1 case of (6.27) is

a 1 bNa 1 cNa 1 dNa 1 eNm b f aNm c f aNm d f aNm e f aN

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m 1Nm f aNa 1N fN

9V8a; b; c; d; e;f; m f N;N; 1

9V8m; a 1 b f; a 1 c f; a 1 d f; a 1 e f;

m f a; m f N;N; 1; 6:29

where m 3a 2 b c d e 2f:This formula is similar to the Bailey transformation formula for terminating

balanced 9F8 series (equivalent to Eq. (2.4.3.1) in [23])

9V8a; b; c; d; e;f; l a N 1 e f;N; 1

a 1Na 1 e fNl 1 eNl 1 fNa 1 eNa 1 fNl 1Nl 1 e fN 9V8l; l b a; l c a; l d a; e;f; l a N 1 e f;N; 1;

6:30

where l 2a 1 b c d: But it is certainly different.

Corollary 6.10. The limiting q-1 case of (6.20) is

XgANn

Dx gDx

Y1pipn

a jgj gi xi xna xi xn

Y1pi; jpn

bj xi xjgi1 xi xjgi

Y1pipnc xi xngie xi xngi

a N 1 xi xngia 1 d xi xngi" # Njgjdjgja 1 cjgja 1 ejgj

Y1pipn

a xi xnjgja 1 bi xi xnjgj

a 1 b1 ? bn cNa 1 b1 ? bn c dNa 1 c dN

a 1 cN

Y1pipna 1 bi d xi xnNa 1 xi xnN

a

1

bi

xi

xn

N

a

1

d

xi

xn

N

6:31

provided 2a N 1 b1 ? bn c d e:

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Remark 6.6. In the case when n 1; (6.31) reduces to Dougalls summation formulafor very-well-poised 7V6 (Eq. (2.3.4.1) in [23])

7V6a; b; c; d; e;N; 1

a 1Na 1 b cNa 1 b dNa 1 c dNa 1 bNa 1 cNa 1 dNa 1 b c dN: 6:32

when 2a N 1 b c d e:

7. Multiple generalization of Sears transformation formula

In this section, we give two types of multiple generalization of Sears

transformation formula for 4f3 balanced basic hypergeometric series

and 2 types of 3f2 terminating basic hypergeometric series and their q-1

limits.

7.1. Multiple Sears transformation

Proposition 7.1.

XgANn

qjgjDxqgDx

Y1pi; jpn

bjxi=xjgiqxi=xjgi

Y1pipn;1pkpm

ckxiyk=xnymgidxiyk=xnymgi

qNjgjajgjejgj fjgj Y1pipn;1pkpm

ckxiyk=xnymgidxiyk=xnymgi

e=aN f=aNeN fNaN

XdANm

qjdjDyqdDy

Y1pk;lpm

d=clyk=yldkqyk=yldk

qNjdjajdj

q1Na=e

jd

jq1Na=f

jd

j Y1pipn;1pkpmd=bixiyk=xnymdk

dxiyk=xnym

dk

7:1

when ab1?bnc1?cmq1N dmef:

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Proof. We consider the product of multiple q-Euler transformation (1.13) and third

Heines transformation (1.9),

XgANn

ujgjDxqgDx

Y1pi; jpn

Ajxi=xjgiqxi=xjgi

(

Y

1pipn;1pkpm

Bkxiyk=xnymgiCxiyk=xnymgi

)

DEu=FNuN

2f1F=D; F=E

F; q; DEu=F

A

1?A

nB

1?B

mu=

Cm

NuN

XdANm

A1?AnB1?Bmu=Cmjdj Dyqd

Dy

(

Y

1pk;lpm

C=Blyk=yldkqyk=yldk

Y1pipn;1pkpm

C=Aixiyk=xnymdkCxiyk=xnymdk

)

2f1D; E

F; q; u

: 7:2

We now suppose that a1?anb1?bm=cm DE=F: By taking the coefficients of the

both side of uN in the equation above, we have

XgANn

qjgjDxqgDx

Y1pi; jpn

Ajxi=xjgiqxi=xjgi

qN

jg

jq1N=F

jg

jq1ND=Fjgjq1NE=FjgjY

1pipn;1pkpm

Bkxiyk=xnym

gi

Cxiyk=xnymgi

DNENF=DNF=ENDE

F

N

XdANm

qjdjDyqdDy

Y1pk;lpm

C=Blyk=yldkqyk=yldk

qNjdjq1N=Fjdj

q1N=Djdjq1N=Ejdj Y1pipn;1pkpmC=Aixiyk=xnymdk

Cxiyk=xnymdk: 7:3

After a suitable change of parameters, we obtain the desired identity. &

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Corollary 7.1. In the case when m 1; (7.1) reduces

XgANn qj

g

jD

xqg

Dx Y1pi; jpnbjxi=xjgiqxi=xjgi

qNjgjajgj

ejgj fjgjY

1pipn

cxi=xngidxi=xngi

e=aN f=aNeN fNaN

n3fn2qN; a; d=b1x1=xn;y; d=bn1xn1=xn; d=bn; d=c

dx1=xn;y; dxn1=xn; d; q1Na=e; q1Na=f;; q; q

: 7:4

Remark 7.1. m n 1 case of this transformation reduces to Sears transformationfor balanced 4f3 series (Eq. (2.10.4) in [5])

4f3qN; a; b; c

d; e;f; q; q

aN e=aN f=aNeN fN 4f3

qN; a; d=b; d=cd; aq1N=e; aq1N=f

; q; q

7:5

when abc defqN1:

We have also another form of multiple Sears transformation.

Proposition 7.2

XgANn

qjgjDxqgDx

Y1pi; jpn

bjxi=xjgiqxi=xjgi

Y1pipn

cxi=xngidxi=xngi

qNjgj

f

jg

j Y1pkpmaym=ykjgj

ekym=yk

jg

j d=cNde1?em=amb1?bncN

Y1pipn

d=bixi=xnNdxi=xnN

Y

1pkpm

aym=ykNekym=ykN

e1?em

am

N

XdANm

qjdjDyqdDy

Y1pk;lpm

el=ayk=yldkqyk=yldk

Y1pkpm

f=ayk=ymdkq1Na1yk=ymdk

qNjdj

q1Nc=djdj Y1pipnq1Nd1xn=xijdj

q1Nbi=dxn=xijdj 7:6

when amb1?bmcq

1N de1?em f:

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Proof. We consider the product of 2 multiple q-Euler transformations of type (4.1)

XgANn

ujgjDxqgDx

Y1pi; jpn

Ajxi=xjgiqxi=xjgi

Y1pipn

Bxi=xngiCxi=xngi

( )

D1?DmEu=FNuN

m1fmF=E; F=D1y1=ym;y;

F=Dm1ym1=ym; F=Dm

Fy1=ym;y;

Fym

1=ym; F; q; D1y; DmEu=F

XdANm

ujdjDyqdDy

Y1pk;lpm

Dlyk=yldkqyk=yldk

Y1pkpm

Eyk=ymdkFyk=ymdk

( )

A1?AnBu=CNuN

n1fnC=B; C=A1x1=xn;y;

C=An1xn1=xn; C=An

Cx1=xn;y;

Cxn1=xn; C; q; A1y; AnBu=C

:

7:7

We now suppose that A1?AnB=C D1?DmE=F: By taking the coefficient ofuNin both sides of (7.7), we obtain

XgANn

qjgjDxqgDx

Y1pi; jpn

Ajxi=xjgiqxi=xjgi

Y1pipn

Bxi=xngiCxi=xngi

qN

jg

jq1NE=FjgjY

1pkpm

q1NF1ym=ykjg

jq1NDk=Fym=ykjgj C=BNF=EN

Y1pipn

C=Aixi=xnNCxi=xnN

Y1pkpm

Fyk=ymNF=Dkyk=ymN

XdANm

qjdjDyqdDy

Y1pk;lpm

Dlyk=yldkqyk=yldk

Y1pkpm

Eyk=ymdkFyk=ymdk

qNjdj

q1NB=Cjdj Y1pipnq1NC1xn=xijdj

q1NAi=Cxn=xijdj: 7:8

By changing of parameters suitably, we get the desired identity. &

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XgANn

qjgjDxqgDx

Y1pi; jpn

bjxi=xjgiqxi=xjgi

qNjgjejgj

ajgj fjgj

Y1pipn

cxi=xngidxi=xngi

d=cNde1?em=amb1?bncNY

1pipn

d=bixi=xnNdxi=xnN

aNeN

e

a

N

n3fn2qN; e=a;f=a; q1Nd1xn=x1;y;

q1Na1; q1Nc=d; q1Nb1=dxn=x1;y;

q1Nd1xn=xn1; q1Nd1

q1Nbn1=dxn=xn1; q1Nbn=d; q; q

: 7:9

Note that right-hand side of (7.9) is essentially the same as that of (7.4).

Remark 7.2. m

n

1 case of (7.6) is itself opposite version of Sears transforma-

tion obtained by reversing the order of summation.

Remark 7.3. Further properties of multiple Sears transformation of type (7.1) is

discussed in [9].

7.2. Multiple 3f2 transformation

In the same way, we obtain two types of multiple transformation formula for 3f2by using the multiple q-Euler transformation formulas and (multiple) q-binomial

theorem.

Proposition 7.3.

XgANnq=d1?dmjgj Dxq

gDx Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y

1pipn

qNjgjq1Ncd1?dm=a1?anbjgj

bxi=xngicxi=xngi

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c=bNcd1?dm=a1?anbNY

1pipn

c=aixi=xnNcxi=xnN

XdANm

qjdj Dyqd

Dy

Y1pk;lpm

dlyk=yldkqyk=yldk

qNjdj

q1Nb=cjdjY

1pipn


: 7:10

Proof. Consider the product of the multiple q-Euler transformation (1.13) and the

multiple q-binomial theorem (1.11)

XgANn

ujgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

( )

d1?dmuNa1?anbu=cN

XdANm

ujdjDyqdDy

Y1pk;lpm

dlyk=yldkqyk=yldk

( )

n1fnc=b;

c=a1

x1=xn;y;

c=an

1

xn

1=xn; c=an

cx1=xn;y; cxn1=xn; c ; q; a1y; anbu=c

: 7:11

By taking the coefficient of uN in the equation above and by using a simple

modification of q-shifted factorials, we have the desired formula. &


XgANn

q=djg

jD

xqg

Dx Y1pi; jpn

ajxi=xj

gi

qxi=xjgi

Y

1pipn

bxi=xngicxi=xngi

" #qNjgj

q1Ncd=a1?anbjgj

c=bNcd=a1?anbNY

1pipn

c=aixi=xnNcxi=xnN

n2fn1qN; d; q1Nc1xn=x1;y;

q1Nb=c; q1N

a1=c

xn=x1;y;q1Nc1xn=xn1; q1Nc1

q1Nan1=cxn=xn1; q1Nan=c; q; q

: 7:12

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Remark 7.4. Eq. (7.10) is a multiple generalization of transformation formula for

terminating 3f2 basic hypergeometric series.

3f2qN; a; b

c; q1Ncd=ab; q; q=d

c=bNcd=abNc=aNcN 3

f2qN; d; q1Nc1

q1Nb=c; q1Na=c; q; q

: 7:13

Proposition 7.4.

XgANn

qjgjDxqgDx Y1pi; jpn

ajxi=xjgiqxi=xjgi Y1pipn;1pkpm


qNjgjq

1

N

d1

jgj a1?anb1?bm=c

mdNdN

XdANm

qjdjDyqdDy

Y1pk;lpm

c=blyk=yldkqyk=yldk

Y1pipn;1pkpm


qNjdj

q1Ncm=a1?anb1?bmdjdj: 7:14

Proof. Multiply the factor uN=d1uN to the both sides of (1.13) and take thecoefficient of un in that equation. &


XgANn

qjgjDxqgDx

Y1pi; jpn

ajxi=xjgiqxi=xjgi

Y1pipn

bxi=xngicxi=xngi

qNjgjq1Nd1jgj

a1?anbd=cN

dN

n2fn1qN; c=b; c=a1x1=xn;y; c=an1xn1=xn; c=an

q1Nc=a1?anbd; cx1=xn;y; cxn1=xn; c; q; q

: 7:15

Remark 7.5. Eq. (7.14) is also a multiple generalization of transformation formula

for terminating 3f2 basic hypergeometric series

3f2qN; a; b

c; q1Nd1; q; q

abd=cNdN 3f2

qN; c=b; c=aq1Nc=abd; c

; q; q

: 7:16

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7.3. Their q-1 limits

Now we write down q-1 limits of the formulas of this section. One can obtain

from multiple Euler transformation (4.8), etc. in the same way as in basichypergeometric case.

Proposition 7.5.

XgANn

Dx gDx

Y1pi; jpn

bj xi xjgi1 xi xjgi

Y1pipn;1pkpmck xi yk xn ymgid xi yk xn ymgi

Njgjajgjejgj fjgj e aN f aNeN fN

XdANm

Dy dDy

Y1pk;lpm

d cl yk yldk1 yk yldk

Y1pipn;1pkpm

d

bi

xi

yk

xn

ym

dk

d xi yk xn ymdk

Njdjajdj1 N a ejdj1 N a fjdj7:17

when a b1 ? bn c1 ? cm 1 N md e f:


XgANn

Dx gDx

Y1pi; jpn

bj xi xjgi1 xi xjgi

Njgjajgjejgj fjgjY

1pipn

c xi xngid xi xngi

e aN f aNeN fNn3Fn2

N; a; d b1 x1 xn;y;d

x1

xn;y;

d bn1 xn1 xn; d bn; d cd xn1 xn; d; 1 N a e; 1 N a f;

; 1

: 7:18

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Remark 7.6. m n 1 case of this transformation reduces to Whipples transfor-mation for balanced 4F3 series

4F3N; a; b; c

d; e;f; 1

e aN f aNeN fN 4F3

N; a; d b; d cd; 1 N a e; 1 N a f ; 1

7:19

when a b c d e f N 1:

We have also another form of the multiple Whipple transformation.

Proposition 7.6.

XgANn

Dx gDx

Y1pi; jpn

bj xi xjgi1 xi xjgi

Y1pipn

c xi xngid xi xngi

Njgj fjgjY

1pkpm

a ym ykjgjekym=ykjgj

d cN

d e1 ? em ma b1 ? bn cN

Y1pipn

d bi xi xnNd xi xnN

Y1pkpm

a ym ykNek ym ykN

XdANm

Dy dDy

Y1pk;lpm

el a yk yldk1 yk yldk

Y

1pkpm

f a yk ymdk1 N a yk ymdk

N

jdj1 N c djdj

Y1pipn

1

N

d

xn

xi

jdj1 N bi d xn xijdj 7:20

when ma b1 ? bm c 1 N d e1 ? em f:


XgANnDx gDx Y1pi; jpn

bj xi xjgi1 xi xjgi Y1pipn

c xi xngid xi xngi

Njgj fjgjajgjejgj

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d cNd e1 ? em ma b1 ? bn cN

aNeN

Y1pipn

d

b

i x

i x

nNd xi xnN

n3Fn2N; e a;f a; 1 N d xn x1;y;

1 N a; 1 N c d; 1 N b1 d xn x1;y;

1 N d xn xn1; 1 N d

1 N bn1 d xn xn1; 1 N bn d; 1

: 7:21

Note that right-hand side of (7.21) is essentially the same as that of (7.18).

Remark 7.7. m n 1 case of (7.20) is itself is opposite version of the Whippletransformation (7.19) obtained by reversing the order of the summation.

In the same way, we obtain two types of multiple transformation formula for 3F2by using the multiple Euler transformation formulas and (multiple) binomial

theorem.

Proposition 7.7

XgANn

Dx gDx

Y1pi; jpn

aj xi xjgi1 xi xjgi

Y1pipn

b xi xngic xi xngi

Njgj1 N c d1 ? dm a1 ? an bjgj c bNc d1 ? dm a1 ? an bN

Y1pipn

c ai xi xnNc xi xnN

XdANmD

y

d

Dy Y1pk;lpmdl yk yldk1 yk yldk

Njdj1 N b cjdjY

1pipn

1 N c xn xijdj1 N ai c xn xijdj

: 7:22



aj xi xjgi1 xi xjgi

Y

1pipn

b xi xngic xi xngi

Njgj1 N c d a1 ? an bjgj

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c bNc d a1 ? an bNY

1pipn

c ai xi xnNc xi xnN

n2Fn1 N; d; 1 N c xn x1;y;1 N b c; 1 N a1 c xn x1;y;

1 N c xn xn1; 1 N c

1 N an1 c xn xn1; 1 N an c; 1

: 7:23

Remark 7.8. Eq. (7.22) is a multiple generalization of transformation formula for

terminating 3F2 basic hypergeometric series.

3F2 N; a; b

c; 1 N c d a b ; 1 c bNc d a bN

c aNcN 3

F2N; d; 1 N c

1 N b c; 1 N a c ; 1

: 7:24

Proposition 7.8.


aj xi xjgi1 xi xjgi

Njgj1 N djgjY

1pipn;1pkpm

bk xi yk xn ymgic xi yk xn ymgi

a1 ? an b1 ? bm mc dNdN

XdANm

Dy dDy

Y1pk;lpm


Y1pipn;1pkpmc ai xi yk xn ymdk

c xi yk xn ymdk Njdj1 N mc a1 ? an b1 ? bm djdj

: 7:25



aj xi xjgi1 xi xjgi

Njgj1 N djgjY

1pipn

b xi xngic xi xngi

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a1 ? an b d cNdN

n2Fn1 N; c

b; c

a

1 x

1 x

n;y;

1 N c a1 ? an b d; c x1 xn;y;

c an1 xn1 xn; c anc xn1 xn; c

; 1

: 7:26

Remark 7.9. Eq. (7.25) is also a multiple generalization of transformation formula

for terminating 3F2 basic hypergeometric series

3F2 N; a; b

c; 1 N d; 1 a b d cNdN 3

F2N; c b; c a

1 N c a b d; c ; 1

: 7:27

Acknowledgments

It is the authors pleasure to thank the referee for valuable comments. He would

express his sincere gratitude to Prof. C. Krattenthaler, Prof. S.C. Milne and Dr. M.Schlosser for useful comments and especially to his adviser Prof. M. Noumi for so

many encouragements, advises and suggestions. He also thanks Dr. K. Iohara and

Dr. H. Rosengren for careful reading of the earlier version of this paper and pointing

out several misprints.

Note added

After finishing this work, H. Rosengren kindly informed the author that he

recovered the results of Section 6 of this paper in [21] by using his reduction formula

of KarlssonMinton type. The author would like to thank again Dr. Rosengren for

informing his result.

References

[1] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.

[2] G.E. Andrews, Summation and transformation for basic Appell series, J. London Math. Soc. 4 (2)

(1972) 618622.

[3] G.E. Andrews, Problems and prospectives for basic hypergeometric functions, in: Theory and

application of special functions, Proc. Advanced Sem. Math. Res. Center, University of Wisconsin,

Madison, WI, 1975, pp. 191224.

[4] R.Y. Denis, R.A. Gustafson, An SU

n

q-beta integral transformation and multiple hypergeometric

series identities, SIAM J. Math. Anal. 23 (2) (1992) 552561.

[5] G. Gasper, M. Rahman, Basic hypergeometric series, in: G.C. Rota (Ed.), Encyclopedia of

Mathematics and Its Applications, Vol. 35. Cambridge University press, Cambridge, 1990.

ARTICLE IN PRESS



45/45

[6] R.A. Gustafson, C. Krattenthaler, Determinant evaluations and Un extension of Heines 2f1-transformations, in: Special functions, q-series and related topics Toronto, On, 1995, Fields Inst.

Commun. 14 (1997) 8389.

[7] W.J. Holman, Summation theorems for hypergeometric series in Un; SIAM J. Math. Anal. 11(1980) 523532.[8] W.J. Holman, L.C. Biedenharn, J.D. Louck, On hypergeometric series well-poised in SUn; SIAM J.

Math. Anal. 7 (1976) 529541.

[9] Y. Kajihara, Some remarks on multiple Sears transformations, Contemp. Math. 291 (2001) 139145.

[10] Y. Kajihara, M. Noumi, Raising operators of row type for Macdonald polynomials, Compositio

Math. 120 (2000) 119136.

[11] A.N. Kirillov, M. Noumi, q-difference raising operators for Macdonald Polynomials and the

integrality of transition coefficients, in: Algebraic Methods and q-Special functions, CRM Proc.

Lecture Notes 22 (1999) 227243.

[12] I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd Edition, Oxford University Press,

Oxford, 1995.

[13] K. Mimachi, M. Noumi, An integral representation of eigenfunction for Macdonalds q-differenceoperators, Tohoku Math. J. 91 (1997) 517525.

[14] S.C. Milne, An elementary proof of Macdonald identities for A1l ; Adv. in Math. 57 (1985) 3470.

[15] S.C. Milne, A q-analogue of hypergeometric series well-poised in SUn and invariant G-functions,Adv. in Math. 58 (1985) 160.

[16] S.C. Milne, A q-analogue of the Gauss summation theorem for hypergeometric series in Un; Adv. inMath. 72 (1988) 59131.

[17] S.C. Milne, A q-analogue of a Whipples transformation of hypergeometric series in Un; Adv. inMath. 108 (1994) 176.

[18] S.C. Milne, Balanced 3f2 summation formulas for Un basic hypergeometric series, Adv. in Math.131 (1997) 93187.

[19] S.C. Milne, G.M. Lily, Consequences of the Al and Cl Bailey transform and Bailey lemma, Discrete

Math. 139 (1995) 319346.

[20] S.C. Milne, J.W. Newcomb, Un very-well-poised 10f9 transformations, J. Comput. Appl. Math. 68(1,2) (1996) 239285.

[21] H. Rosengren, Reduction formula for KarlssonMinton type hypergeometric series, Constr. Approx.

to appear.

[22] M. Schlosser, Summation theorems for multidimensional basic hypergeometric series by determinant

evaluations, Discrete Math. 210 (2000) 151169.

[23] L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.

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