Parameter Augmentation for Basic Hypergeometric Series, II

Transcription

1 journal of combinatorial theory Series A (997) article no TA97280 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 People's Republic of China and Zhi-Guo Liu Department of Mathematics Xinxiang Education College Xinxiang Henan People's Republic of China Communicated by the Managing Editors Received February 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 997 Academic Press INTRODUCTION Since the umbral calculus was developed by G-C Rota and his collaborators [30 3] there has been extensive interest in an operator approach to basic hypergeometric series as in the work of Goldman and Rota [6 7] 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 * chent7lanlgov and chensunnankaieducn Copyright 997 by Academic Press All rights of reproduction in any form reserved

2 76 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 [2] 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 summation formula the q-pfaffsaalschu tz 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 [5] Let q < and the q-shifted factorial be defined by (a; q) 0 = Clearly n& (a; q) n = ` (&aq k ) (a; q) = ` (&aq k ) () (a; q) n =(a; q) (aq n ; q) (2) Throughout we shall adopt the following notation of multiple q-shifted factorials (a a 2 a m ; q) n =(a ; q) n (a 2 ; q) n }}}(a m ;q) n (a a 2 a m ; q) =(a ; q) (a 2 ; q) }}}(a m ;q) The q-binomial coefficient is defined by _ n k& = (q; q) k &k The basic hypergeometric series r+ r is defined by r+ r\ a a r+ ; q x b b += (a a r+ ; q) n x n r (q b b r ; q) n In this paper we will frequently use the Cauchy identity and its special cases [5]

3 BASIC HYPERGEOMETRIC SERIES 77 (ax; q) (a; q) n x n = (3) (x; q) = (x; q) (&x; q) = x n (4) q (n 2 ) x n (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) (2) a By convention D 0 q 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] D n q [f(a) g(a)]= n q k(k&n) _ n k& Dk q [f(a)]dn&k q [g(q k a)] (22) The following property of D q is straightforward but important Theorem 2 D q{ Dq{ k (at; q) = = t (23) (at; q) (at; q) = = tk (24) (at; q) The operator to be used in this paper denoted T is constructed based on D q T(bD q )= (bd q ) n (25) From the above theorem we may obtain the following

4 78 CHEN AND LIU Theorem 22 T(bD q ) { (at; q) = = (at bt; q) (26) Employing the Leibniz formula we may derive the following theorem Theorem 23 Applying the Leibniz formula for D q the left- Proof of Theorem 23 hand side of (27) equals as desired b n D n q{ = = T(bD q ) { (as at; q) = = (abst; q) (27) (as at bs bt; q) b n (as at; q) = b n n q k(k&n) n q k(k&n) _ n k& q{ Dk _ n s k& k (at; q) k (bs) k = (as at; q) (q; q) k (abst; q) = (as at; q) (bs; q) (bt; q) = (abst; q) (as at bs bt; q) K (as; q) = Dn&k (tq k ) n&k (as; q) (atq k ; q) n=k (bt) n&k &k q { (atq k ; q) = Theorem 23 reduces to Theorem 22 when s=0 These two theorems may lead to many important results in the theory of hypergeometric series 3 THE q-pfaffsaalschu TZ 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-pfaffsaalschu tz formula the Gauss 2 summation formula and the Jackson q-analogue of

5 BASIC HYPERGEOMETRIC SERIES 79 the Euler transform Using the idea of parameter augmentation the q-pfaff Saalschu tz formula can easily be derived from the q-chuvandermonde convolution formula the Gauss 2 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 [3]) 2 \ q&n Using the following relation a (ca; q) n ; q q (3) c +=an (c; q) n (3) can be written as n a n (ca; q) n =(&c) n q (n 2 ) (aq &n c; q) (aqc; q) (32) (q &n ; q) k q k q ( n 2) =(&c) n (33) (q; q) k (c; q) k (aq k aq &n c; q) (c; q) n (aqc a; q) Applying the operator T(bD q ) on both sides of (33) it follows that n Using the relations T(bD q ) { (q &n ; q) k q k (q c; q) k T(bD q ) { (aq k aq &n c; q) = = (&c)n q ( n 2 ) (c; q) n T(bD q ) { (aqc a; q) = (aq k aq &n c; q) = = (abq &n+k c; q) (aq k aq &n c bq k bq &n c; q) T(bD q ) { (aqc a; q) = = (abqc; q) (aqc a bqc b; q) we obtain the q-pfaffsaalschu tz formula [ ] Theorem 3 a b q&n 3 2\ c abc & q &n; q q + =(ca cb; q) n (34) (c cab; q) n

6 80 CHEN AND LIU Next we proceed to give an augmentation argument for the q-gauss theorem By (2) the Cauchy binomial theorem can be written as = (x; q) (a; q) x n (aq n ax; q) Applying T(bD q ) on both sides of the above identity we get (x; q) T(bD q ) { (a; q) = = Since and T(bD q ) { it follows from (35) that x n T(bD q ) { (aq n ax; q) = (35) T(bD q ) { (a; q) = = (a b; q) (aq n ax; q) = = (abxq n ; q) (aq n ax bq n bx; q) x n (abxq n ; q) = (x a b; q) (aq n ax bq n bx; q) Using (2) the above identity can be written as 2 \ a b abx ; q x + =(ax bx; q) (x abx; q) which is equivalent to the following form of the q-gauss theorem [ ] Theorem 32 2 \ a b c ; q c ab+ =(ca cb; q) (36) (c cab; q) By the Cauchy binomial theorem we have (abx; q) = (x; q) (abx; q) = (b; q) (ab; q) n x n (ax; q) n b n

7 BASIC HYPERGEOMETRIC SERIES 8 It follows that which can be rewritten as (ab; q) n x n = (b; q) (x; q) x n = (b; q) (abq n ax; q) (x; q) Applying T(cD q ) we have (ax; q) n b n b n (axq n ab; q) which can be rewritten as x n (abcxq n ; q) (abq n ax bcq n cx; q) = (b; q) b n (abcxq n ; q) (x; q) (q; q) n (axq n ab cxq n bc; q) ab bc 2 \ abcx ; q x + =(b; q) ax cx 2 (x; q) \ abcx ; q b + (37) Note that (37) is equivalent to the Jackson q-analogue of the Euler transform [5] Theorem 33 2 \ a b c ; q x + =(abxc; q) 2 (x; q) \ ca cb ; q abx c c + (38) 4 AN IDENTITY OF ANDREWS In [] 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 [ 3] (a; q) n q (n+ 2 ) =(&q; q) (aq; q 2 ) (4)

8 82 CHEN AND LIU Proof The left hand side of (4) equals (a; q) q (n+) 2 =(a; q) (aq n ; q) j=0 =(a; q) j=0 a j =(&q a; q) = (&q a; q) (a; q 2 ) (q; q) j q nj q ( n+) 2 a j (q; q) j (&q j+ ; q) j=0 =(&q; q) (aq; q 2 ) a j (q 2 ; q 2 ) j as desired K Let us rewrite (4) in the following form q (n+ 2 ) (aq n aq n+ ; q 2 ) = (&q; q) (a; q 2 ) Setting q to q 2 the above identity becomes q 2 ( n+) 2 = (&q2 ; q 2 ) (42) (q 2 ; q 2 ) n (aq n2 ; aq (n+)2 ; q) (a; q) Applying the operator T(bD q ) on both sides of (42) leads to the following identity q 2 ( n+ 2 ) (abq n+(2) ; q) (q 2 ; q 2 ) n (aq n2 aq (n+)2 bq n2 bq (n+)2 ; q) = (&q2 ; q 2 ) (a b; q) (43) Finally setting q 2 back to q we get the Andrews identity [3] Theorem 4 (a b; q) n q (n+) 2 = (&q; q) (aq bq; q 2 ) (abq; q 2 ) n (abq; q 2 ) (44)

9 BASIC HYPERGEOMETRIC SERIES 83 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 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 + n=& where $ n m is the Kronecker delta By the Cauchy binomial identity we have = (ae i% ae &i% ; q) (&) n q (n) 2 _ N+n& 2N =$ 0 N (5) = n + = k=& a n e in% a m e &im% (q; q) m a m+n (q; q) m e i(n&m)% e &ik% From the Jacobi triple product identity we have a 2m&k (q; q) m&k (q; q) m (52) (e 2i% e &2i% ; q) =(&e &2i% )(e 2i% qe 2i% ; q) = + (&) n q (n) 2 (&e &2i% ) e 2ni% (q; q) n=& = + (&) n (+q n ) q (n) 2 e 2ni% (53) (q; q) n=&

10 84 CHEN AND LIU Thus we obtain? &? (e 2i% e &2i% ; q) (ae i% ae &i% ; q) d% = + (q; q) n=& a 2m&k + k=& (&) n q (n 2 ) (+q n ) _ (q; q) m&k (q; q) m? e (2n&k) i% d% &? The following identity is straightforward? e (2n&k) i% d%=2?$ k2n &? Hence we have the following evaluation? &? (e 2i% e &2i% ; q) (ae i% ae &i% ; q) d% a 2m&2n = 2? + (&) n (+q n ) q (n) 2 (q; q) (q; q) n=& m&2n (q; q) m = 2? a 2N + (&) n (+q n ) q (n) 2 (q; q) (q; q) N=0 2N _ n+n& 2N n=& = 4? a 2N + (&) n q (n 2) (q; q) (q; q) N=0 2N _ n+n& 2N n=& = 4? a 2N $ N0 (q; q) (q; q) N=0 2N = 4? (q;q) Since the above integral is an even function of % we obtain the following theorem Theorem 5? 0 (e 2i% e &2i% ; q) d%= 2? (54) (ae i% ae &i% ; q) (q; q) 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

11 BASIC HYPERGEOMETRIC SERIES 85 augmentation on the above special case By adding the parameter b to (54) one obtains the identity which is closely related to the orthogonality of the RogersSzego polynomials [ ] 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 [5] h(cos %; r)=(re i% re &i% ; q) h(cos %; a a 2 a m )=h(cos %; a ) h(cos %; a 2 )}}}h(cos %; a m ) Hence (54) becomes? 0 h(cos 2%;) h(cos %; a) d%= 2? (q; q) (55) Taking the action of T(bD q ) on both sides of (55) we obtain? 0 h(cos 2%; )T(bD q ) { 2? d%= h(cos %; a)= (q; q) where T(bD q ) { h(cos %; a)= =T(bD q) { (ae i% ae &i% ; q) = (ab; q) = (ae i% ae &i% be i% be &i% ; q) = (ab; q) h(cos %; a b) It follows that? 0 h(cos 2%; ) 2? d%= (56) h(cos %; a b) (q ab; q) Taking the action of T(cD q ) on both sides of (56) we obtain? 0 h(cos 2%; ) h(cos %; b) T(cD q) { h(cos %; a)= d% = 2? (q; q) T(cD q ) { (ab; q) =

12 86 CHEN AND LIU where Thus we have T(cD q ) { h(cos %; a)= = (ac; q) h(cos %; a c) T(cD q ) { (ab; q) = = (ab bc; q)? 0 h(cos 2%; ) h(cos %; a b c) d%= 2? (57) (q ab ac bc; q) One more action of T(dD q ) on the above identity leads to the Askey Wilson integral where? 0 h(cos 2%;) h(cos %; b c) T(dD q) { h(cos %; a)= d% = 2? (q bc; q) T(dD q ) { (ab ac; q) = (58) T(dD q ) { h(cos %; a)= = (ad; q) h(cos %; a d) T(dD q ) { (ab ac; q) = = (abcd; q) (ab ac bd cd; q) The above identity (58) is just the AskeyWilson integral [ ] Theorem 52 (AskeyWilson)? 0 h(cos 2%; ) h(cos %; a b c d) d%= 2?(abcd; q) (59) (q ab ac ad bc bd cd; q) where max[ a b c d ]< 52 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 [9] Here we give a proof in a few lines

13 BASIC HYPERGEOMETRIC SERIES 87 Proof of the IsmailStantonViennot Integral T( fd q ) on the AskeyWilson integral one obtains Taking the action of? 0 h(cos 2%; ) h(cos %; a b c d f ) d% 2? = T( fd q ) { (abcd; q) (q af bc bd cd; q) (ab ac ad; q) = (50) By the Leibniz formula it follows that T( fd q ) { (abcd; q) (ab ac ad; q) = = = (dc; q) n b n f k a D k (q; q) k q{ n (ac ad; q) = f k k (dc; q) n b n (q; q) k j=0 _ { (ac ad; q) = Dk& j q (aq j ) n (dc; q) n b n ( fd q ) j = (q; q) n (q; q) j=0 j { = = = (dc; q) n b n n (bf ) m (q; q) m a n&m (bf ) m (q; q) m j( j&k) q _ k j& D j q q j(n&m) a n&m n (ac ad; q) = _ m& n f m T( fq n&m D q ) { (dc; q) k+m (q; q) k (ab) k T( fq k D q ) { (ac ad; q) = ac ad; q) = (dc; q) k+m (q; q) k (ab) k (acdfq k ; q) (ac ad fcq k fdq k ; q) = (acdf; q) (cf df cd; q) k (ab) k (ac ad cf df; q) (q acdf; q) k _ n m& f m (cdq k ; q) m (bf ) m (q; q) m = (acdf; q) (cf df cd; q) k (ab) k (bcdfqk ; q) (ac ad cf df; q) (q acdf; q) k (bf; q) = (acdf bcdf; q) cf df cd (ac ad bf cf df; q) 3 2\ acdf bcdf ; q ab + (5) Combining (50) and (5) we are led to the IsmailStantonViennot integral [9]

14 88 CHEN AND LIU Theorem 53? 0 h(cos 2%; ) h(cos %; a b c d f ) d% 2?(acdf bcdf; q) cf df cd = (q ac ad af bc bd bf cd cf df; q) 3 2\ acdf bcdf ; q ab + (52) where max[ a b c d f ]< 53 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-pfaffsaalschu tz formula Replacing f by fq n in (52) multiplying (abcd) n (q abcdf 2 ; q) n on both sides and taking summation over n we get? 0 h(cos 2%;) h(cos %; a b c d f ) \ fei% fe &i% ; q abcd d% (53) 2 abcdf + 2 2?(acdf bcdf; q) = (q ac ad af bc bd bf cd cf df; q) _ 3 2\ (cf df cd; q) n (q acdf bcdf; q) n (ab) n q &n af bf abcdf 2 q &n cd ; q q + (54) By the q-gauss summation formula (36) we have 2 \ fei% fe &i% abcdf ; q abcd h(cos %; abcdf ) = (55) + 2 (abcd abcdf 2 ; q) By the q-pfaffsaalschu tz formula (34) we have q 3 2\ &n af bf abcdf 2 q &n cd ; q q + = (bcdf acdf; q) n (56) (abcdf 2 cd; q) n

15 BASIC HYPERGEOMETRIC SERIES 89 Substituting (55) and (56) into (54) it follows that? h(cos 2%; )h(cos %; abcdf ) d% 0 h(cos %; a b c d f ) = 2?(acdf bcdf abcd abcdf 2 ; q) cf df (q ac ad af bc bd bf cd cf df; q) 2 \ abcdf 2; q ab + (57) Applying the q-gauss summation formula again we have cf df 2 \ abcdf 2; q ab + = (abdf abcf; q) (58) (abcdf 2 ab; q) Substituting (58) into (57) we are led to the NassrallahRahman integral [23] Theorem 54? 0 h(cos 2%; )h(cos %; abcdf ) h(cos %; a b c d f ) d% where max[ a b c d f ]< = 2?(abcd abcf abdf acdf bcdf; q) (q ab ac ad af bc bd bf cd cf df; q) (59) 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 [5] in the light of parameter augmentation The following is the AndrewsAskey integral which can be derived from Ramanujan's sum of Theorem 6 d c (qtc qtd; q) d (at bt; q) q t= d(&q)(q dqc cd abcd; q) (6) (ac ad bc bd; q) Dividing both sides of (6) by (abcd; q) we obtain d c (qtc qtd; q) (at bt abcd; q) d q t= d(&q)(q dqc cd; q) (ac ad bc bd; q)

16 90 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 ) { (at abcd; q) = = (abcdet; q) (at et abcd ebcd; q) T(eD q ) { (ca da; q) = = (acde; q) (ca da ce de; q) Theorem 62 d c (qtc qtd abcdet; q) d q t (at bt et; q) = d(&q)(q dqc cd abcd bcde acde; q) (ac ad bc bd ce de; q) (62) 7 AN EXTENSION OF THE ASKEYROY INTEGRAL GASPER'S FORMULA We observe that a recent integral formula discovered by Gasper [4] 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 2?? &? (\e i% d qde &i% \ \ce &i% qe i% c\; q) (ae i% be i% ce &i% de &i% ; q) d% = (abcd \cd dq\c \ q\; q) (q ac ad bc bd; q) (7) Dividing both sides of the above equation by (abcd; q) we have 2?? &? (\e i% d qde &i% \ \ce &i% qe i% c\; q) (ae i% be i% ce &i% de &i% abcd; q) d% = (\cd dq\c \ q\; q) (q ac ad bc bc bd; q) (72)

17 BASIC HYPERGEOMETRIC SERIES 9 Taking the action of T( fd q ) on both sides of (72) and applying the relations we obtain T( fd q ) { (ae i% abcd; q) = = (abcdfe i% ; q) (ae i% fe i% abcd bcdf; q) T( fd q ) { (ac ad; q) = = (acdf; q) (ac ad cf df; q) 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) (73) where max[ a b c d ]< cd\{0 This is exactly the formula recently discovered by Gasper [4] 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 8 9] 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 [ ] The RogersSzego polynomial is defined by which has the following generating function n h n (x q)= _ n k& xk (8) h n (x q) t n = (82) (t xt; q) The q-hermite polynomials H n (x q) is often defined by its generating function [] t n H n (x q) = ` (&2xtq n +t 2 q 2n )

18 92 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 (83) Using the above operator definition of the RogersSzego polynomial and our augmentation argument it is easy to derive Mehler's formula Theorem 8 t n h n (x q) h n (y q) = (xyt2 ; q) (84) (t xt yt xyt; q) Proof t n h n (x q) h n (y q) = =T(D q ) { h n (y q) T(D q ) { (xt)n = h n (y q) (xt)n = =T(D q ) { (xt xyt; q) = = (xyt2 ; 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 [] based the recurrence relation As is shown below the Rogers formula becomes apparent from the point view of parameter augmentation

19 BASIC HYPERGEOMETRIC SERIES 93 Theorem 82 t n s m h m+n (x q) (q; q) m =(tsx; q) t n h n (x q) h m (x q) s m (q; q) m (85) Proof The left-hand side of (85) equals t n s m T(D q ) x m+n (q; q) m =T(D q ) =T(D q ) (tx sx; q) (tx) n (sx) m (q; q) m =(tsx; q) (tx sx t s; q) =(tsx; q) (t tx; q) (s sx; q) =(tsx; q) t n h n (x q) h m (x q) s m (q; q) m as required K Substituting the expansion = (tsx; q) (ts) n x n in (85) and comparing the coefficients of t n s m we get the linearization formula for h n (x q) Theorem 83 min[m n] h n (x q) h m (x q)= _ n k&_ m k& (q; q) k x k h n+m&2k (x q) (86)

20 94 CHEN AND LIU A combinatorial proof of the above formula (86) is given by Ismail Stanton and Viennot [9] Similarly using the expansion (tsx; q) = (&) n q (n) 2 x n t n s n in (85) we are led to the inverse relation of the linearization formula obtained by Askey and Ismail [8] Theorem 84 min[m n] h m+n (x q)= _ n k&_ m k& (q; q) k q ( k 2 ) (&x) k h n&k (x q) h m&k (x q) (87) ACKNOWLEDGMENTS This work was done under the auspices of the US 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 G E Andrews q-identities of Auluck Carlitz and Rogers Duke Math J 33 (966) G E Andrews On the foundation of combinatorial theory V Eulerian differential operators Stud Appl Math 50 (97) G E Andrews Applications of basic hypergeometric functions SIAM Rev 6 (974) G E Andrews The Theory of Partitions in ``Encylopedia of Mathematics and Its Applications'' Vol 2 AddisonWesley Reading MA 976 [reissued by Cambridge Univ Press Cambridge 985] 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 G E Andrews and R Askey Another q-extension of the beta function Proc Amer Math Soc 8 (98) R Askey An elementary evalution of a beta type integral Indian J Pure Appl Math 4 (983) R Askey and M E H Ismail A generalization of ultraspherical polynomials in ``Studies in Pure Mathematics'' (P Erdo s Ed) pp 5578 Birkha user Boston MA R Askey and J Wilson Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials Memoirs Amer Math Soc 54 No 39 (985) 0 N M Atakishiyev and Sh M Nagiyev On the RogersSzego polynomials J Phys A Math Gen 27 (994) L66L65

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