Parameter Augmentation for Basic Hypergeometric Series, II

Journal of Combinatorial Theory, Series A  TA2801 journal of combinatorial theory, Series A 80, 175195 (1997) article no. TA972801

Parameter Augmentation for Basic Hypergeometric Series, II William Y. C. Chen* T-7, Mail Stop B284, Los Alamos National Laboratory, Los Alamos, New Mexico 87545; and Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China

and Zhi-Guo Liu Department of Mathematics, Xinxiang Education College, Xinxiang, Henan 453000, People's Republic of China Communicated by the Managing Editors Received February 27, 1997

In a previous paper, we explored the idea of parameter augmentation for basic hypergeometric series, which provides a method of proving q-summation and integral formula based special cases obtained by reducing some parameters to zero. In the present paper, we shall mainly deal with parameter augmentation for q-integrals such as the AskeyWilson integral, the NassrallahRahman integral, the q-integral form of Sears transformation, and Gasper's formula of the extension of the AskeyRoy integral. The parameter augmentation is realized by another operator, which leads to considerable simplications of some well known q-summation and transformation formulas. A brief treatment of the RogersSzego polynomials is also given.  1997 Academic Press

1. INTRODUCTION Since the umbral calculus was developed by G.-C. Rota and his collaborators [30, 31], there has been extensive interest in an operator approach to basic hypergeometric series, as in the work of Goldman and Rota [16, 17], Andrews [2], and Roman [29]. Instead of aiming at a general approach to polynomials of q-binomial type, we focus our attention on some specific operators which can be simply defined as certain exponential operators. It turns out that such simple operators play a fundamental * E-mail: chent7.lanl.gov and chensun.nankai.edu.cn.

175 0097-316597 25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

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CHEN AND LIU

role in the theory of basic hypergeometric series. In the present paper we continue the study of parameter augmentation based on an exponential operator which is dual to the operator introduced in a previous paper [12]. By using an augmentation operator an identity on basic hypergeometric series with multiple parameters may be recovered from its special case obtained by setting some parameters to zero. Many important results on q-summations and q-integrals naturally fall into the framework of this augmentation operator such as the AskeyWilson integral, the Nassrallah Rahman integral, the q-integral form of Sears transformation, Gasper's formula of the extension of the AskeyRoy integral, the q-Gauss 2 , 1 summation formula, the q-PfaffSaalschutz formula, Jackson's q-analogue of the Euler transform, some identities on RogersSzego polynomials, or their equivalent forms for the q-Hermite polynomials, and an identity of Andrews generalizing the Lebesgue identity. We shall follow the notation and terminology in [15]. Let |q| <1 and the q-shifted factorial be defined by n&1



(a; q) n = ` (1&aq k ),

(a; q) 0 =1,

(a; q)  = ` (1&aq k ).

k=0

(1.1)

k=0

Clearly, (a; q) n =(a; q)  (aq n ; q)  .

(1.2)

Throughout we shall adopt the following notation of multiple q-shifted factorials: (a 1 , a 2 , ..., a m ; q) n =(a 1 ; q) n(a 2 ; q) n } } } (a m ; q) n , (a 1 , a 2 , ..., a m ; q)  =(a 1 ; q) (a 2 ; q)  } } } (a m ; q)  . The q-binomial coefficient is defined by (q; q) n

n

_k& = (q; q) (q; q) k

The basic hypergeometric series

r+1

,r

\

r+1

. n&k

, r is defined by

 a 1 , ..., a r+1 (a 1 , ..., a r+1 ; q) n x n ; q, x = : . b 1 , ..., b r (q, b 1 , ..., b r ; q) n n=0

+

In this paper, we will frequently use the Cauchy identity and its special cases [15]:

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177

BASIC HYPERGEOMETRIC SERIES  (ax; q)  (a; q) n x n =: , (x; q)  n=0 (q; q) n

(1.3)

 xn 1 =: , (x; q)  n=0 (q; q) n

(1.4)

n

q( 2 ) x n . (&x ; q)  = : (q; q) n n=0 

(1.5)

2. THE EXPONENTIAL OPERATOR T (bD q ) The usual q-differential operator, or q-derivative, is defined by D q f (a)=

f (a)& f (aq) . a

(2.1)

By convention, D 0q is understood as the identity. The Leibniz rule for D q is the following identity, which is a variation of the q-binomial theorem [29]: n

D nq[ f (a) g(a)]= : q k(k&n) k=0

n

_k& D [ f(a)] D k q

n&k q

[ g(q ka)].

(2.2)

The following property of D q is straightforward, but important: Theorem 2.1. t

1

{(at; q) = =(at; q) t 1 = D { (at ; q) = (at; q) Dq



,

(2.3)

.

(2.4)



k

k q





The operator to be used in this paper, denoted T, is constructed based on D q : 

(bD q ) n . (q; q) n n=0

T (bD q )= :

From the above theorem, we may obtain the following.

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(2.5)

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Theorem 2.2. T (bD q )

{

1 1 = . (at; q)  (at, bt; q) 

=

(2.6)

Employing the Leibniz formula, we may derive the following theorem. Theorem 2.3. T (bD q )

(abst; q) 

1

{ (as, at; q) = = (as, at, bs, bt; q) 

.

(2.7)



Proof of Theorem 2.3. Applying the Leibniz formula for D q , the lefthand side of (2.7) equals 

bn 1 D nq (q; q) n (as, at; q)  n=0

{

:

=



n bn n 1 1 : q k(k&n) D kq D n&k q k (q; q) (as; q) (atq ; q)  k n k=0  n=0

=: 

n bn : q k(k&n) (q; q) n k=0 n=0

=:

_& { n s _k& (as; q) k

=

{

=

(tq k ) n&k k  (atq ; q) 

=

 1 (at; q) k (bs) k  (bt) n&k : : (as, at; q)  k=0 (q; q) k (q; q) n&k n=k

=

1 (abst; q)  1 (as, at; q)  (bs; q)  (bt; q) 

=

(abst; q)  , (as, at, bs, bt; q) 

as desired. K Theorem 2.3 reduces to Theorem 2.2 when s=0. These two theorems may lead to many important results in the theory of hypergeometric series.

3. THE q-PFAFFSAALSCHUTZ FORMULA AND THE JACKSON TRANSFORM To give the reader a taste of the applications of the operator T, we will present probably the simplest proofs of the well-known q-PfaffSaalschutz formula, the Gauss 2 , 1 summation formula, and the Jackson q-analogue of

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the Euler transform. Using the idea of parameter augmentation, the q-Pfaff Saalschutz formula can easily be derived from the q-ChuVandermonde convolution formula, the Gauss 2 , 1 summation formula can easily be derived from the Cauchy binomial theorem, and the Jackson q-analogue of the Euler transform can also be recovered from the Cauchy binomial theorem written in a slightly different form. Recall that the usual q-ChuVandermonde convolution can be rewritten as the following basic hypergeometric series form (see, for example, [13]): 2 ,1

\

a n(ca; q) n q &n, a . ; q, q = (c; q) n c

+

(3.1)

Using the following relation n

a n(ca; q) n =(&c) n q( 2 )

(aq 1&nc; q)  , (aqc; q) 

(3.2)

(3.1) can be written as n (q &n ; q) k q k q( 2 ) n =(&c) . : (q; q) k (c; q) k (aq k, aq 1&nc; q)  (c; q) n (aqc, a; q)  k=0 n

(3.3)

Applying the operator T (bD q ) on both sides of (3.3), it follows that n

(q &n ; q) k q k 1 T (bD q ) k 1&n (q, c; q) (aq , aq c; q)  k k=0 n ) ( (&c) n q 2 1 = T (bD q ) . (c; q) n (aqc, a; q) 

{

:

{

=

=

Using the relations 1

(abq 1&n+kc; q)  , (aq k, aq 1&nc, bq k, bq 1&nc; q) 

{ (aq , aq c; q) = (abqc; q) 1 T (bD ) { (aqc, a; q) = = (aqc, a, bqc, b; q)

T (bD q )

k

=

1&n





,

q





we obtain the q-PfaffSaalschutz formula [5, 15, 20, 35]: We have

Theorem 3.1.

3

,2

\

(ca, cb; q) n a, b, q &n ; q, q = . c, abc &1q 1&n (c, cab; q) n

+

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(3.4)

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CHEN AND LIU

Next we proceed to give an augmentation argument for the q-Gauss theorem. By (1.2), the Cauchy binomial theorem can be written as  xn 1 =: . (x; q)  (a; q)  n=0 (q; q) n (aq n, ax; q) 

Applying T (bD q ) on both sides of the above identity, we get  1 1 xn 1 T (bD q ) =: T (bD q ) . n (x; q)  (a; q)  (q; q) n (aq , ax; q)  n=0

{

=

{

=

(3.5)

Since T (bD q )

1

1

{ (a; q) = = (a, b; q) 



and T (bD q )

{

(abxq n ; q)  1 = , n (aq , ax; q)  (aq , ax, bq n, bx; q)  n

=

it follows from (3.5) that  1 xn (abxq n ; q)  =: . (x, a, b; q)  n=0 (q; q) n (aq n, ax, bq n, bx; q) 

Using (1.2), the above identity can be written as 2

,1

\

(ax, bx; q)  a, b , ; q, x = (x, abx; q)  abx

+

which is equivalent to the following form of the q-Gauss theorem [3, 5, 15, 20, 29]: Theorem 3.2.

We have 2

,1

\

(ca, cb; q)  a, b c = . ; q, ab (c, cab; q)  c

+

By the Cauchy binomial theorem, we have  (abx; q)  (ab; q) n n =: x , (x; q)  (q; q) n n=0

(abx; q)   (ax; q) n n =: b. (b; q)  (q; q) n n=0

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(3.6)

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BASIC HYPERGEOMETRIC SERIES

It follows that 

(ab; q) n n (b; q)   (ax; q) n n x = : b , (q; q) n (x; q)  n=0 (q; q) n n=0 :

which can be rewritten as 

(b; q)   xn bn = : . n (q; q) n (abq , ax; q)  (x; q)  n=0 (q; q) n (axq n, ab; q)  n=0 :

Applying T (cD q ), we have 

x n(abcxq n ; q)  (q; q) n (abq n, ax, bcq n, cx; q)  n=0 :

=

(b; q)   b n(abcxq n ; q)  : , (x; q)  n=0 (q; q) n (axq n, ab, cxq n, bc; q) 

which can be rewritten as 2

,1

(b; q) 

ab, bc

\ abcx ; q, x+ = (x; q)

2 

,1

ax, cx

\ abcx ; q, b+ .

(3.7)

Note that (3.7) is equivalent to the Jackson q-analogue of the Euler transform [15]: Theorem 3.3. 2

,1

\

We have (abxc; q)  a, b ca, cb abx . ; q, x = ; q, 2 ,1 (x; q)  c c c

+

\

+

(3.8)

4. AN IDENTITY OF ANDREWS In [1], Andrews gives an identity which contains as a special case the Lebesgue identity. This identity is also called the q-analogue of the Gauss second theorem. Here we point out that the Andrews identity can be recovered from the Lebesgue identity by parameter augmentation. To this end, we present a proof of the Lebesgue identity for completeness. It is stated as follows [1, 3]: ( n+1 ) (a; q) n q 2 =(&q; q)  (aq; q 2 )  . : (q; q) n n=0 

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(4.1)

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CHEN AND LIU

Proof.

The left hand side of (4.1) equals

( n+1 ) ( n+1 )  q 2 1 a j  q nj q 2 (a; q)  : =(a; q)  : : n (q; q) n n=0 (q; q) n (aq ; q)  j=0 (q; q) j n=0 



aj (&q j+1 ; q)  (q; q) j j=0

=(a; q)  :



aj (q 2 ; q 2 ) j j=0

=(&q, a; q)  : =

(&q, a; q)  (a; q 2 ) 

=(&q; q)  (aq; q 2 )  , as desired. K Let us rewrite (4.1) in the following form: ( n+1 ) (&q; q)  q 2 : = . n n+1 2 (q; q) (aq , aq ; q ) (a; q 2 )  n  n=0 

Setting q to q 12, the above identity becomes ( n+1 ) (&q 12 ; q 12 )  q 12 2 = . : (q 12 ; q 12 ) n (aq n2 ; aq (n+1)2 ; q)  (a; q)  n=0 

(4.2)

Applying the operator T (bD q ) on both sides of (4.2) leads to the following identity: ( n+1 ) (&q 12 ; q 12 )  q 12 2 (abq n+(12) ; q)  = . : 12 12 n2 (n+1)2 n2 (n+1)2 ; q ) n (aq , aq , bq , bq ; q)  (a, b; q)  n=0 (q (4.3) 

Finally, setting q 12 back to q, we get the Andrews identity [3]: Theorem 4.1.

We have

( n+1 ) (&q; q)  (aq, bq; q 2 )  (a, b; q) n q 2 = . : (q; q) n (abq; q 2 ) n (abq; q 2 )  n=0 

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(4.4)

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BASIC HYPERGEOMETRIC SERIES

5. THE ASKEYWILSON INTEGRAL AND THE NASSRALLAHRAHMAN INTEGRAL In this section, we shall present a treatment of the AskeyWilson integrals via parameter augmentation. The point of departure from the usual AskeyWilson integral is the orthogonality relation obtained from the Cauchy binomial theorem and the Jacobi triple product identity. Once we have the identity with one parameter, while the AskeyWilson integral involves four parameters, the augmentation turns out to be an easy task. Moreover, one more augmenation argument on the AskeyWilson integral leads to a generalization due to IsmailStantonViennot. We also point out that the NassrallahRahman integral can be easily derived from the result of IsmailStantonViennot. 5.1. The AskeyWilson Integral In the Cauchy binomial theorem, setting a=q &2N, x=q N, where N is an nonnegative integer, one obtains the following orthogonality relation: +

(&1) n q

:

( n2 )

n=&

2N

_N+n& =$

0, N

,

(5.1)

where $ n, m is the Kronecker delta. By the Cauchy binomial identity we have  a ne in%  a me &im% 1 =: : &i% (ae , ae ; q)  n=0 (q ; q) n m=0 (q; q) m i%



a m+n e i(n&m) % (q; q) (q; q) n m n, m=0

= :



+

a 2m&k . (q; q) m&k (q; q) m m=0

e &ik% :

= : k=&

(5.2)

From the Jacobi triple product identity, we have (e 2i%, e &2i%; q)  =(1&e &2i% )(e 2i%, qe 2i%; q)  = =

1 (q; q)  1 (q; q) 

+

:

(&1) n q

( n2 )

(1&e &2i% ) e 2ni%

n= & +

:

n

(&1) n (1+q n ) q( 2 )e 2ni%.

n= &

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(5.3)

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CHEN AND LIU

Thus, we obtain (e 2i%, e &2i%; q)  d% i% &i% ; q)  &? (ae , ae

|

?

=

+ 1 : (q; q)  n= &

_



+

: (&1) n q

:

( n2 )

(1+q n )

k=& m=0

a 2m&k (q; q) m&k (q; q) m

|

?

e (2n&k) i% d%.

&?

The following identity is straightforward:

|

?

e (2n&k) i% d%=2?$ k, 2n .

&?

Hence we have the following evaluation: (e 2i%, e &2i%; q)  d% i% &i% ; q)  &? (ae , ae

|

?

= = =

2? (q; q) 

+

:



: (&1) n (1+q n ) q

( n2 )

n= & m=0

 2? a 2N : (q; q)  N=0 (q; q) 2N

n= &



+

2N

4? a : (q; q)  N=0 (q; q) 2N 

a 2m&2n (q; q) m&2n (q; q) m

+

: : n= &

(&1) n (1+q n ) q n

(&1) n q( 2 )

( n2 )

2N

_n+N&

2N

_n+N&

2N

=

4? a : $ N, 0 (q; q)  N=0 (q; q) 2N

=

4? . (q; q) 

Since the above integral is an even function of %, we obtain the following theorem. Theorem 5.1.

We have

|

?

0

2? (e 2i%, e &2i%; q)  d%= . (ae i%, ae &i%; q)  (q; q) 

(5.4)

The above identity is a very special case of the AskeyWilson integral with other parameters b, c, d reduced to zero. Our aim is to arrive at the general case of the AskeyWilson integral by three steps of parameter

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BASIC HYPERGEOMETRIC SERIES

augmentation on the above special case. By adding the parameter b to (5.4) one obtains the identity which is closely related to the orthogonality of the RogersSzego polynomials [10, 19, 32, 33]. These polynomials are a variant form of the q-Hermite polynomials. In the last section, we will revisit some fundamental properties of RogersSzego polynomials by the approach of parameter augmentation. For notational simplicity, we adopt the following notation [15]: h(cos %; r)=(re i%, re &i%; q)  , h(cos %; a 1 , a 2 , ..., a m )=h(cos %; a 1 ) h(cos %; a 2 ) } } } h(cos %; a m ). Hence (5.4) becomes

|

?

0

2? h(cos 2%; 1) d%= . h(cos %; a) (q; q) 

(5.5)

Taking the action of T(bD q ) on both sides of (5.5), we obtain

|

?

h(cos 2%; 1) T(bD q )

0

{

2? 1 d%= , h(cos %; a) (q; q) 

=

where 1

1

{ h(cos %; a)= =T(bD ) {(ae , ae

T(bD q )

q

i%

&i%

; q) 

=

=

(ab; q)  (ae i%, ae &i%, be i%, be &i%; q) 

=

(ab; q)  . h(cos %; a, b)

It follows that

|

?

0

2? h(cos 2%; 1) d%= . h(cos %; a, b) (q, ab; q) 

Taking the action of T(cD q ) on both sides of (5.6), we obtain

|

? 0

h(cos 2%; 1) 1 T(cD q ) d% h(cos %; b) h(cos %; a) =

2? (q; q) 

{ = 1 T(cD ) { (ab; q) = , q



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(5.6)

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CHEN AND LIU

where (ac; q) 

1

{h(cos %; a)= =h(cos %; a, c) , 1 1 T(cD ) {(ab; q) = =(ab, bc; q) .

T(cD q )

q





Thus, we have

|

?

0

2? h(cos 2%; 1) d%= . h(cos %; a, b, c) (q, ab, ac, bc; q) 

(5.7)

One more action of T(dD q ) on the above identity leads to the Askey Wilson integral,

|

?

0

h(cos 2%; 1) 1 T(dD q ) d% h(cos %; b, c) h(cos %; a)

{

=

=

2? 1 T(dD q ) , (q, bc; q)  (ab, ac; q) 

{

=

(5.8)

where (ad; q) 

1

{h(cos %; a)= =h(cos %; a, d ) , (abcd; q) 1 T(dD ) { (ab, ac; q) = = (ab, ac, bd, cd; q) T(dD q )



.

q





The above identity (5.8) is just the AskeyWilson integral [7, 9, 18, 19, 21, 25, 34]: Theorem 5.2. (AskeyWilson). We have

|

?

0

2?(abcd; q)  h(cos 2%; 1) d%= , h(cos %; a, b, c, d ) (q, ab, ac, ad, bc, bd, cd; q) 

(5.9)

where max[|a|, |b|, |c|, |d | ]<1. 5.2. The IsmailStantonViennot Integral One naturally wonders what would happen if one tried to add another parameter, say f, to the AskeyWilson integral. It is interesting that such a consideration of parameter augmentation on the AskeyWilson integral leads to an integral formula obtained by IsmailStantonViennot [19]. Here we give a proof in a few lines:

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BASIC HYPERGEOMETRIC SERIES

Proof of the IsmailStantonViennot Integral. Taking the action of T( fD q ) on the AskeyWilson integral, one obtains

|

?

0

h(cos 2%; 1) d% h(cos %; a, b, c, d, f ) =

2? (abcd; q)  T( fD q ) . (q, af, bc, bd, cd; q)  (ab, ac, ad; q) 

{

=

(5.10)

By the Leibniz formula, it follows that T( fD q )

(abcd; q) 

{ (ab, ac, ad; q) = 



n

(dc; q) n b  fk an : D kq (q; q) n k=0 (q; q) k (ac, ad; q)  n=0

{

=: 

=

k (dc; q) n b n  fk k : : q j( j&k) D qj (q; q) n k=0 (q; q) k j=0 j n=0

_&

=: _

{

1 j D k& (aq j ) n q (ac, ad; q) 

=



(dc; q) n b n  ( fD q ) j 1 : (q; q) (q; q) (ac, ad; q)  n j n=0 j=0

=: 

(dc; q) n b (q; q) n n=0

n

n

: a n&m

=:



m

m=0

{ n _ m& f



(bf ) (dc; q) k+m : (q; q) m k=0 (q; q) k m=0

= :

n

_ m& f 1 T( fq D ) {(ac, ad; q) = 1 (ab) T( fq D ) {ac, ad; q) = m

=

n

: q j(n&m)a n&m

m

m=0

n&m

q



k

k

q





(bf ) m  (dc; q) k+m (acdfq k; q)  : (ab) k (q; q) m k=0 (q; q) k (ac, ad, fcq k, fdq k; q)  m=0

= : =

  (acdf; q)  (cf, df, cd; q) k (cdq k; q) m : (ab) k : (bf ) m (ac, ad, cf, df; q)  k=0 (q, acdf; q) k (q; q) m m=0

=

 (acdf; q)  (cf, df, cd; q) k (bcdfq k; q)  : (ab) k (ac, ad, cf, df; q)  k=0 (q, acdf; q) k (bf; q) 

=

(acdf, bcdf; q)  cf, df, cd ; q, ab . 3,2 (ac, ad, bf, cf, df; q)  acdf, bcdf

\

+

(5.11)

Combining (5.10) and (5.11), we are led to the IsmailStantonViennot integral [19]:

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188

CHEN AND LIU

We have

Theorem 5.3.

|

?

0

h(cos 2%; 1) d% h(cos %; a, b, c, d, f ) =

cf, df, cd 2?(acdf, bcdf; q)  ; q, ab , 3, 2 (q, ac, ad, af, bc, bd, bf, cd, cf, df; q)  acdf, bcdf

\

+

(5.12) where max[|a|, |b|, |c|, |d |, | f | ]<1. 5.3. The NassrallahRahman Integral We point out that the well-known NassrallahRahman integral originally obtained by the integral representation of the Sears transformation and the Bailey 8 , 7 transformation can easily be derived from the above Ismail StantonViennot integral by an application of the q-Gauss summation formula and the q-PfaffSaalschutz formula. Replacing f by fq n in (5.12), multiplying (abcd ) n (q, abcdf 2; q) n on both sides, and taking summation over n, we get

|

?

0

h(cos 2%; 1) fe i%, fe &i% , ; q, abcd d% 2 1 h(cos %; a, b, c, d, f ) abcdf 2

\

=

+

(5.13)

 (cf, df, cd; q) n 2?(acdf, bcdf; q)  : (ab) n (q, ac, ad, af, bc, bd, bf, cd, cf, df; q)  n=0 (q, acdf, bcdf; q) n

_ 3,2

\

q &n, af, bf ; q, q . abcdf 2, q 1&ncd

+

(5.14)

By the q-Gauss summation formula (3.6), we have 2, 1

\

h(cos %; abcdf ) fe i%, fe &i% ; q, abcd = . 2 abcdf (abcd, abcdf 2; q) 

+

(5.15)

By the q-PfaffSaalschutz formula (3.4), we have

3

,2

\

(bcdf, acdf; q) n q &n, af, bf ; q, q = . (abcdf 2, cd; q) n abcdf 2, q 1&ncd

+

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(5.16)

189

BASIC HYPERGEOMETRIC SERIES

Substituting (5.15) and (5.16) into (5.14), it follows that

|

?

0

h(cos 2%; 1) h(cos %; abcdf ) d% h(cos %; a, b, c, d, f ) =

2?(acdf, bcdf, abcd, abcdf 2; q)  cf, df , ; q, ab . (5.17) (q, ac, ad, af, bc, bd, bf, cd, cf, df; q)  2 1 abcdf 2

\

+

Applying the q-Gauss summation formula again, we have

2

,1

\

(abdf, abcf; q)  cf, df ; q, ab = . 2 (abcdf 2, ab; q)  abcdf

+

(5.18)

Substituting (5.18) into (5.17), we are led to the NassrallahRahman integral [23]: Theorem 5.4.

|

? 0

We have h(cos 2%; 1) h(cos %; abcdf ) d% h(cos %; a, b, c, d, f ) =

2?(abcd, abcf, abdf, acdf, bcdf; q)  , (q, ab, ac, ad, af, bc, bd, bf, cd, cf, df; q) 

(5.19)

where max[|a|, |b|, |c|, |d |, | f | ]<1.

6. THE q-INTEGRAL FORM OF THE SEARS TRANSFORMATION In this section, we point out the relationship between the AndrewsAskey integral [6] and the q-integral form of the Sears transformation [15] in the light of parameter augmentation. The following is the AndrewsAskey integral which can be derived from Ramanujan's sum of 1  1 : Theorem 6.1.

|

d

c

We have

d(1&q)(q, dqc, cd, abcd; q)  (qtc, qtd; q)  d q t= . (at, bt; q)  (ac, ad, bc, bd; q) 

Dividing both sides of (6.1) by (abcd; q)  , we obtain

|

d

c

d(1&q)(q, dqc, cd; q)  (qtc, qtd; q)  d q t= . (at, bt, abcd; q)  (ac, ad, bc, bd; q) 

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(6.1)

190

CHEN AND LIU

Taking the action of T(eD q ) on both sides of the above identity, the q-integral form of Sears transformation can be immediately obtained from the following relations: T(eD q )

{

T(eD q )

Theorem 6.2.

|

d c

(abcdet; q)  1 = , (at, abcd; q)  (at, et, abcd, ebcd; q)  1

{(ca, da; q)

= (acde; q) = = (ca, da, ce, de; q) 



. 

We have

(qtc, qtd, abcdet; q)  dq t (at, bt, et; q)  =

d(1&q)(q, dqc, cd, abcd, bcde, acde; q)  . (ac, ad, bc, bd, ce, de; q) 

(6.2)

7. AN EXTENSION OF THE ASKEYROY INTEGRAL: GASPER'S FORMULA We observe that a recent integral formula discovered by Gasper [14] and proved also by Rahman and Suslov [26] can be derived from the AskeyRoy integral in one step of parameter augmentation. The Askey Roy integral [26] is given by 1 2?

|

(\e i%d, qde &i%\, \ce &i%, qe i%c\; q)  d% (ae i%, be i%, ce &i%, de &i%; q)  &? ?

=

(abcd, \cd, dq\c, \, q\; q)  . (q, ac, ad, bc, bd; q) 

(7.1)

Dividing both sides of the above equation by (abcd; q)  , we have 1 2?

|

(\e i%d, qde &i%\, \ce &i%, qe i%c\; q)  d% (ae i%, be i%, ce &i%, de &i%, abcd; q)  &? ?

=

( \cd, dq\c, \, q\; q)  . (q, ac, ad, bc, bc, bd; q) 

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(7.2)

191

BASIC HYPERGEOMETRIC SERIES

Taking the action of T( fD q ) on both sides of (7.2), and applying the relations T( fD q )

{

(abcdfe i%; q)  1 = , (ae i%, abcd; q)  (ae i%, fe i%, abcd, bcdf; q) 

T( fD q )

{

1 (ac, ad; q) 

= (acdf; q) = = (ac, ad, cf, df; q) 

, 

we obtain 1 2?

|

(\e i%d, qde &i%\, \ce &i%, qe i%c\, abcdfe i%; q)  d% (ae i%, be i%, fe i%, ce &i%, de &i%; q)  &? ?

=

(\cd, dq\c, \, q\, abcd, bcdf, acdf; q)  , (q, ac, ad, bc, bd, cf, df; q) 

(7.3)

where max[ |a|, |b|, |c|, |d | ]<1, cd\{0. This is exactly the formula recently discovered by Gasper [14].

8. THE ROGERSSZEGO POLYNOMIALS The RogersSzego polynomials play an important role in the theory of orthogonal polynomials, particularly in the study of the AskeyWilson integral [8, 18, 19]. We observe that some important results on the Rogers Szego polynomials or the equivalent forms on q-Hermite polynomials naturally fall into the framework of parameter augmentation such as Mehler's formula, and the linearization formula and its inverse [4, 8, 11, 18, 19, 22]. The RogersSzego polynomial is defined by n

h n(x | q)= : k=0

n k x , k

_&

(8.1)

which has the following generating function: 

: h n(x | q) n=0

tn 1 = . (q; q) n (t, xt; q) 

(8.2)

The q-Hermite polynomials H n(x | q) is often defined by its generating function [11]: 

: H n(x | q) n=0

 tn 1 . =` (q; q) n n=0 (1&2xtq n +t 2q 2n )

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CHEN AND LIU

The RogersSzego polynomials and the q-Hermite polynomials are related by H n(cos % | q)=e &in%h n(e 2i% | q). The polynomials h n (x | q) can easily be represented by the augmentation operator as follows: h n(x | q)=T(D q ) x n.

(8.3)

Using the above operator definition of the RogersSzego polynomial and our augmentation argument, it is easy to derive Mehler's formula. Theorem 8.1.

We have



: h n(x | q) h n( y | q) n=0

tn (xyt 2; q)  = . (q; q) n (t, xt, yt, xyt; q) 

(8.4)

Proof. 

: h n(x | q) h n( y | q) n=0

 tn (xt) n = : h n( y | q) T(D q ) (q; q) n n=0 (q; q) n

{

=T(D q )

=T(D q )

{



: h n( y | q) n=0

(xt) n (q; q) n

= =

1

{(xt, xyt; q) = 

=

(xyt 2; q)  , (t, xt, yt, xyt; q) 

as desired. K The above formula is a special case of the following identity due to Rogers [27, 28] which implies the linearization formula for RogersSzego polynomials, and accordingly for the q-Hermite polynomials. A simple proof of the Rogers formula is given by Bressoud [11] based the recurrence relation. As is shown below, the Rogers formula becomes apparent from the point view of parameter augmentation,.

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BASIC HYPERGEOMETRIC SERIES

Theorem 8.2.

We have





:

: h m+n(x | q)

n=0 m=0

tn sm (q; q) n (q; q) m





=(tsx; q)  :

: h n(x | q) h m(x | q)

n=0 m=0

Proof.

193

tn sm . (q; q) n (q; q) m

(8.5)

The left-hand side of (8.5) equals 



:

: T(D q ) x m+n

tn sm (q; q) n (q; q) m

n=0 m=0 



(tx) n (sx) m (q; q) n (q; q) m m=0

=T(D q ) :

:

n=0

=T(D q )

1 (tx, sx; q) 

=(tsx; q)  (tx, sx, t, s; q)  =(tsx; q) 

1 1 (t, tx; q)  (s, sx; q)  



=(tsx; q)  :

: h n(x | q) h m(x | q)

n=0 m=0

tn sm , (q; q) n (q; q) m

as required. K Substituting the expansion  1 (ts) n x n =: (tsx; q)  n=0 (q; q) n

in (8.5) and comparing the coefficients of t ns m, we get the linearization formula for h n(x | q): Theorem 8.3.

We have min[m, n]

h n(x | q) h m(x | q)=

: k=0

n k

m (q; q) k x kh n+m&2k(x | q). (8.6) k

_ &_ &

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194

CHEN AND LIU

A combinatorial proof of the above formula (8.6) is given by Ismail, Stanton, and Viennot [19]. Similarly, using the expansion ( n) (&1) n q 2 n n n x ts (tsx; q)  = : (q; q) n n=0 

in (8.5), we are led to the inverse relation of the linearization formula obtained by Askey and Ismail [8]: Theorem 8.4.

We have

min[m, n]

h m+n(x | q)=

: k=0

n k

m ( k) (q; q) k q 2 (&x) k h n&k(x | q) h m&k(x | q). k (8.7)

_ &_ &

ACKNOWLEDGMENTS This work was done under the auspices of the U.S. Department of Energy, the National Science Foundation of China and the Stroock Foundation. The authors thanks G. E. Andrews, W. C. Chu, Q. H. Hou, M. E. H. Ismail, J. D. Louck, and G.-C. Rota for helpful discussions. We are also grateful to the referee for many valuable suggestions.

REFERENCES 1. G. E. Andrews, q-identities of Auluck, Carlitz and Rogers, Duke Math. J. 33 (1966), 575581. 2. G. E. Andrews, On the foundation of combinatorial theory V, Eulerian differential operators, Stud. Appl. Math. 50 (1971), 345375. 3. G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441484. 4. G. E. Andrews, The Theory of Partitions, in ``Encylopedia of Mathematics and Its Applications,'' Vol. 2, AddisonWesley, Reading, MA, 1976 [reissued by Cambridge Univ. Press, Cambridge, 1985]. 5. G. E. Andrews, ``q-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra,'' CBMS Regional Conference Lecture Series, Vol. 66, Amer. Math. Soc, Providences, RI, 1986. 6. G. E. Andrews and R. Askey, Another q-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981), 97100. 7. R. Askey, An elementary evalution of a beta type integral, Indian J. Pure. Appl. Math. 14 (1983), 892895. 8. R. Askey and M. E. H. Ismail, A generalization of ultraspherical polynomials, in ``Studies in Pure Mathematics'' (P. Erdos, Ed.), pp. 5578, Birkhauser, Boston, MA, 1983. 9. R. Askey and J. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs. Amer. Math. Soc. 54, No. 319 (1985). 10. N. M. Atakishiyev and Sh. M. Nagiyev, On the RogersSzego polynomials, J. Phys. A Math. Gen. 27 (1994), L661L615.

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195

11. D. M. Bressoud, A simple proof of Mehler's formula for q-Hermite polynomials, Indiana Univ. Math. J. 29 (1980), 577580. 12. W. Y. C. Chen and Z. G. Liu, Parameter augmentation for basic hypergeometric series, I, in ``Proc. Rotafest,'' Birkhauser, Boston, to appear. 13. W. C. Chu, Basic hypergeometric identities: An introductory revisiting through the Carlitz inversions, Forum Math. 7 (1995), 117129. 14. G. Gasper, q-Extensions of Barnes', Cauchy's, and Euler's beta integrals, in ``Topics in Mathematical Analysis'' (T. M. Rassias, Ed.), pp. 294314, World Scientific, Singapore, 1989. 15. G. Gasper and M. Rahman, ``Basic Hypergeometric Series,'' Cambridge Univ. Press, Cambridge, MA, 1990. 16. J. Goldman and G.-C. Rota, The number of subspaces of a vector space, in ``Recent Progress in Combinatorics'' (W. Tutte, Ed.), pp. 7583, Academic Press, New York, 1969. 17. J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions, Stud. Appl. Math. 49 (1970), 239258. 18. M. E. H. Ismail and D. Stanton, On the AskeyWilson and Rogers polynomials, Canad. J. Math. 40 (1988), 10251045. 19. M. E. H. Ismail, D. Stanton, and G. Viennot, The combinatorics of q-Hermite polynomials and the AskeyWilson integral, Europ. J. Combin. 8 (1987), 379392. 20. J. T. Joichi and D. Stanton, Bijective proofs of basic hypergeomentric series identities, Pacific J. Math. 127 (1987), 103120. 21. E. G. Kalnins and W. Miller, Symmetry techniques for q-series: AskeyWilson polynamials, Rocky Mountain J. Math. 19 (1989), 223240. 22. C. Markett, Linearization of the product of symmetric orthogonal polynomials, Constr. Approx. 10 (1994), 317338. 23. B. Nassrallah and R. Rahman, Projection formulas, a reproducing kernel and a generating function for q-Wilson polynomials, SIAM J. Math. Soc. 16 (1985), 211229. 24. P. Paule, On identities of the Rogers-Ramanujan type, J. Math. Anal. Appl. 107 (1985), 255284. 25. M. Rahman, A simple evaluation of Askey and Wilson's q-integral, Proc. Amer. Math. Soc. 92 (1984), 413417. 26. M. Rahman and S. K. Suslov, Barnes and Ramanujan-Type integrals on the q-linear lattice, SIAM J. Math. Anal. 25 (1994), 10021022. 27. L. J. Rogers, On a three-fold symmetry in the elements of Heine's series, Proc. London Math. Soc. 24 (1893), 171179. 28. L. J. Rogers, On the expansion of some infinite products, Proc. London Math. Soc. 24 (1893), 337352. 29. S. Roman, More on the umbral calculus, with Emphasis on the q-umbral caculus, J. Math. Anal. Appl. 107 (1985), 222254. 30. S. Roman and G.-C. Rota, The umbal calculus, Adv. in Math. 27 (1978), 95188. 31. G.-C. Rota, ``Finite Operator Calculus,'' Academic Press, New York, 1975. 32. G. Szego, Ein Beitrag zur Theorie der Theta funktionen, Sitz. Preuss. Akad. Wiss. Phus. Math. KL. 19 (1926), 242252 [Collected Works, Vol. 1, pp. 795805, Birkhauser, Boston]. 33. G. Szego, ``Orthogonal Polynomials,'' Amer. Math. Soc, Colloquium publications, Vol. 23, 4th ed., American Math. Society, Providence, RI, 1975. 34. H. S. Wilf and Zeilberger, An algorithmic Proof theory for hypergeometric (ordinary and ``q'') multisumintegral identities, Invent. Math. 108 (1992), 575633. 35. D. Zeilberger, A q-Foata proof of the q-Saalschutz identity, Europ. J. Combin. 8 (1987), 461463.

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