A new identity for the sum of products of generalized basic hypergeometric functions
aa r X i v : . [ m a t h . C A ] S e p A NEW IDENTITY FOR THE SUM OF PRODUCTS OF GENERALIZEDBASIC HYPERGEOMETRIC FUNCTIONS
S.I. KALMYKOV, D. KARP, AND A. KUZNETSOV
To the memory of Richard Askey
Abstract.
We present a q -extension of the duality relation for the generalized hypergeometricfunctions established recently by the second and the third named authors which also generalizesthe q -hypergeometric identity due to the third named author (jointly with Feng and Yang). Thisduality relation has the form of a reduction formula for a sum of products of basic hypergeometricfunctions r φ r − . We further derive a confluent version of our formula for t φ r − with t < r , whichinvolves a mixture of two types of basic hypergeometric functions. Our identity is closely relatedto some recent results due to Yamaguchi on contiguous relations for φ basic hypergeometricfunction. Keywords: basic hypergeometric function, basic hypergeometric identity, duality relation, residuetheorem
MSC2010: 33D15 1.
Introduction
Basic hypergeometric series are nearly as old as the standard ones with certain particular casesconsidered already by Euler and Jacobi. The general case was introduced by Heine in 1846 about30 years after Gauss presented the ordinary hypergeometric series. Naturally, over these 174 yearsof history, a huge number of identities in the form of transformation and summation formulaswere discovered with deep connections to number theory, orthogonal polynomials, mathematicalphysics and many other fields. The standard reference is the book by Gasper and Rahman [7].Another comprehensive treatment can be found in [5], while more accessible introduction is in[8]. A nice collection [11] reflects many aspects of more modern developments.The purpose of this paper is to present a q -analogue of the ordinary hypergeometric identitydiscovered recently in [10] by the second and the third named authors. This identity reducesa sum of products of the generalized hypergeometric functions to a Laurent polynomial (withexplicit lower and upper degrees) times a power of (1 − z ) . Its particular cases were discoveredpreviously by Ebisu [4] as an ingredient in calculation of the coefficients of contiguous relationsfor the Gauss hypergeometric function F , by the third named author jointly with Feng andYang in [6] and by the first and second named authors in [9]. Another identity of this type wasdeduced by Beukers and Jouhet in [3], but only a particular case of their formula reduces to ouridentity from [10], while in their full generality these results are independent. The work by Feng,Kuznetsov and Yang also a contains a q -extension [6, Theorem 2]. More recently, Yamaguchi[14] extended the results of Ebisu to q -case by calculating the coefficients of contiguous relationsfor the basic hypergeometric series φ (see definition (5) below). A by-product of his resultis an identity for a quadratic form composed of φ functions [14, Lemma 3] . Our identityproved in Theorem 1 below generalizes both Feng, Kuznetsov and Yang formula obtained bysetting n i = 0 , i = 1 , . . . , r , and Yamaguchi’s formula obtained by setting r = 2 , applying Date : October 1, 2020.First author supported by NSFC grant 11901384.Research of A.K. supported by the Natural Sciences and Engineering Research Council of Canada.
Heine’s transformation and renaming parameters, as explained in the remark following the proofof Theorem 1. We further present a confluent form of our formula in Theorem 2. Finally, wenote that another identity for quadratic form of the basic hypergeometric series was establishedin 2015 in [13, Theorem 1.1]. 2.
Main results
Before we turn to our results let us introduce some notation and definitions. We will use thedefinition of the q -shifted factorial valid for both positive and negative values of the index ([7,(1.2.15) and (1.2.28)]):(1) ( a ; q ) = 1 , ( a ; q ) n = n − Y k =0 (1 − aq k ) , ( a ; q ) − n = n Y k =1 − a/q k ) , n ∈ N . It is easy to see that ( q a ; q ) n = ( − n q an + n ( n − / ( q − a ; q ) − n (2) = ( − n q an + n ( n − / ( q − a ; 1 /q ) n = ( − n ( q − a − n ; q ) n q an + n ( n − / . This definition works for any complex a and q . If we restrict q to | q | < , we can also define ( a ; q ) ∞ = lim n →∞ ( a ; q ) n , where the limit can be shown to exist as a finite number for all complex a . The q -gamma functionis given by [7, (1.10.1)], [8, (21.16)](3) Γ q ( z ) = (1 − q ) − z ( q ; q ) ∞ ( q z ; q ) ∞ for | q | < and all complex z such that q z + k = 1 for all k ∈ N . Comparing this definition with(1) we immediately see that(4) Γ q ( z + k )Γ q ( z ) = ( q z ; q ) k (1 − q ) k for both positive and negative k .Suppose t, s are positive integers, a = ( a , . . . , a t ) ∈ C t , b = ( b , . . . , b s ) ∈ C s . The symbol a [ j ] (where ≤ j ≤ t ) stands for the vector a with j -th component omitted: a [ j ] = ( a , . . . , a j − , a j +1 , . . . , a t ) , and, by definition, a + β = ( a + β, . . . , a t + β ) . For n = ( n , . . . , n t ) ∈ Z t we use the following abbreviations for the products: ( a ) n = ( a ; q ) n ( a ; q ) n · · · ( a t ; q ) n t and Γ q ( a ) = Γ q ( a ) · · · Γ q ( a t ) . We will further write q a = ( q a , . . . , q a t ) . The basic hypergeometric function is defined as follows[7, formula (1.2.22)](5) t φ s (cid:18) ab q, z (cid:19) = ∞ X n =0 ( a ; q ) n ( a ; q ) n · · · ( a t ; q ) n ( b ; q ) n ( b ; q ) n · · · ( b s ; q ) n ( q ; q ) n h ( − n q ( n ) i s − t z n , where t ≤ s + 1 , and the series converges for all z if t ≤ s and for | z | < if t = s + 1 [7,section 1.2]. NEW IDENTITY FOR THE SUM OF PRODUCTS OF GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS3
We will also need the version of the basic hypergeometric function considered by Bailey [2,section 8.1] and Slater [12, (3.2.1.11)] which is defined as follows(6) t ˆ φ s (cid:18) ab q, z (cid:19) = ∞ X n =0 ( a ; q ) n ( a ; q ) n · · · ( a t ; q ) n ( b ; q ) n ( b ; q ) n · · · ( b s ; q ) n ( q ; q ) n z n . Observe [7, p.5] that the series (5) has the property that if we replace z by z/b t and let b t → ∞ ,then the resulting series is again of the form (5) with t replaced by t − (this is known as theconfluence process ). At the same time the Bailey and Slater series (6) can be obtained from the t = s + 1 case of (5) by setting some of the parameters equal to zero:(7) t ˆ φ s (cid:18) a , . . . , a t b , . . . , b s q, z (cid:19) = s +1 φ s (cid:18) a , . . . , a t , , . . . , b , . . . , b s q, z (cid:19) . Note also that in the case t = s +1 the functions t φ s and t ˆ φ s coincide. Since the base q is the samein all formulas in this paper, in what follows we omit it from the notation of the q -hypergeometricfunctions.Our main result is the following Theorem 1.
Assume that q is a complex number satisfying | q | < , a ∈ C r satisfies a i − a j / ∈ Z for ≤ i < j ≤ r and b ∈ C r is arbitrary; assume further that m , n ∈ Z r . Then, for somecomplex numbers β j , the following identity holds (8) r X i =1 q a i ( q − b + a i ; q ) m − n i z − n i ( q a i − a [ i ] ; q ) n [ i ] − n i +1 r φ r − (cid:18) q b − a i q a [ i ] − a i W z (cid:19) r φ r − (cid:18) q − b + a i + m − n i q − a [ i ] + a i + n [ i ] − n i z (cid:19) = 1( W z ; q ) p +1 p − m min X k = − n max β k z k , where m min = min ≤ i ≤ r ( m i ) , n max = max ≤ i ≤ r ( n i ) , M = r P i =1 m i , N = r P i =1 n i , p = max {− , M − N − r +1 } and W = q r − r Y i =1 q a i − b i . Proof.
Using the Cauchy product we expand(9) S = r X i =1 q a i ( q − b + a i ; q ) m − n j z − n i ( q a i − a [ i ] ; q ) n [ i ] − n i +1 r φ r − (cid:18) q b − a i q a [ i ] − a i W z (cid:19) r φ r − (cid:18) q − b + a i + m − n i q − a [ i ] + a i + n [ i ] − n i z (cid:19) = r X i =1 ( q − b + a i ; q ) m − n i z − n i ( q a i − a [ i ] ; q ) n [ i ] − n i +1 ∞ X k =0 z k k X j =0 q a i ( q b − a i ; q ) j ( q − b + a i + m − n i ; q ) k − j ( q a − b q r − ) j ( q a [ i ] − a i ; q ) j ( q − a [ i ] + a i + n [ i ] − n i ; q ) k − j ( q ; q ) j ( q ; q ) k − j = r X i =1 ∞ X k =0 z k − n i k X j =0 q a i ( q − b + a i ; q ) m − n i ( q b − a i ; q ) j ( q − b + a i + m − n i ; q ) k − j ( q a − b q r − ) j ( q a i − a [ i ] ; q ) n [ i ] − n i +1 ( q a [ i ] − a i ; q ) j ( q − a [ i ] + a i + n [ i ] − n i ; q ) k − j ( q ; q ) j ( q ; q ) k − j = r X i =1 ∞ X k =0 z k − n i k X j =0 γ ki,j = r X i =1 ∞ X k i = − n i z k i k i + n i X j =0 γ k i + n i i,j , where we denoted(10) γ ki,j = q a i ( q − b + a i ; q ) m − n i ( q b − a i ; q ) j ( q − b + a i + m − n i ; q ) k − j ( q a − b q r − ) j ( q a i − a [ i ] ; q ) n [ i ] − n i +1 ( q a [ i ] − a i ; q ) j ( q − a [ i ] + a i + n [ i ] − n i ; q ) k − j ( q ; q ) j ( q ; q ) k − j . S.I. KALMYKOV, D. KARP, AND A. KUZNETSOV
Using (2) and (4) we obtain for each term in the numerator in the right-hand side in (10) ( q − b l + a i ; q ) m l − n i ( q b l − a i ; q ) j ( q − b l + a i + m l − n i ; q ) k − j = ( q b l − a i ; q ) j (1 − q ) m l − n i + k − j Γ q (1 − b l + a i + m l − n i )Γ q (1 − b l + a i ) Γ q (1 − b l + a i + m l − n i + k − j )Γ q (1 − b l + a i + m l − n i )= ( q b l − a i ; q ) j ( q − b l + a i ; q ) m l − n i + k − j = ( − j q ( b l − a i ) j + j ( j − / ( q − b l + a i − j ; q ) j ( q − b l + a i ; q ) m l − n i + k − j = ( − j q ( b l − a i ) j + j ( j − / ( q − b l + a i − j ; q ) m l − n i + k . After a similar calculation for denominator terms in (10), we arrive at(11) γ ki,j = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k − n i ( q a i − a [ i ] − j ; q ) n [ i ] + k − n i +1 ( q ; q ) j ( q ; q ) k − j , which is equivalent to(12) γ k + n i i,j = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k ( q a i − a [ i ] − j ; q ) n [ i ] + k +1 ( q ; q ) j ( q ; q ) k + n i − j . According to (2), we have ( q ; q ) j = ∞ for j < so that also ( q ; q ) k − j = ∞ for j > k and thusdefinition (10) implies that γ k + n i i,j = 0 for j < and j > k + n i . These relations extend thedefinition of γ k + n i i,j to arbitrary j , and, in view of this convention, we have by rearranging (9): S = ∞ X k = − n max z k r X i =1 k + n i X j =0 γ k + n i i,j . Next, note that k + m i ≥ for all i = 1 , . . . , r if k ≥ − m min . If − m min ≤ − n max , then k ≥ − m min for all terms in the above sum. Otherwise if − m min > − n max we can write S ( z ) = − m min − X k = − n max α k z k + ∞ X k = − m min z k r X i =1 k + n i X j =0 γ k + n i i,j = − m min − X k = − n max α k z k + S ( z ) , where α k = r X i =1 k + n i X j =0 γ k + n i i,j . Next, for each k ∈ Z define the function f k ( z ) = − ( zq − b ; q ) k + m ( zq − a ; q ) k + n +1 . According to (1), these functions are rational for all k ∈ Z . Further, if k ≥ − m min the expressionin the numerator is a polynomial and all poles come from the zeros of the denominator. If k + n i + 1 > for all i = 1 , . . . , r , then the poles of f k ( z ) are at the points: ( zq − a ; q ) k + n +1 = 0 ⇔ z = q a i − j , i = 1 , . . . , r and j = 0 , . . . , k + n i . Our assumption a i − a j / ∈ Z for ≤ i < j ≤ r guarantees that all poles are simple. Thus, aftersimple calculation, we obtain res z = q ai − j f k ( z ) = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k ( q a i − a [ i ] − j ; q ) n [ i ] + k +1 ( q ; q ) j ( q ; q ) k + n i − j = γ k + n i i,j . If k + n i + 1 ≤ for some values of k and i , then ( zq − a ; q ) k + n i +1 has no zeros at the points z = q a i − j , j = k + n i , . . . , , so that these points do not contribute to the sum of residues of NEW IDENTITY FOR THE SUM OF PRODUCTS OF GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS5 f k ( z ) . In view of the above comment on the values of γ k + n i i,j for j < , this implies that for k ≥ − m min X over all poles of f k ( z ) res f k ( z ) = r X i =1 k + n i X j =0 γ k + n i i,j , where only the terms with k + n i ≥ are non-vanishing. Next, we use the fact that the sum ofresidues of a rational function at all finite points equals its residue at infinity [1, (4.1.14)], whichis the coefficient at z − in the asymptotic expansion f k ( z ) = M − N − r X j = −∞ C j ( k ) z j . Hence, r X i =1 k + n i X j =0 γ k + n i i,j = C − ( k ) . Our key observation is the following lemma.
Lemma 1.
For z large enough we have (13) f k ( z ) = A k z M − N − r X l ≥ Q l ( q − k ) z − l , where A = r Y i =1 q a i − b i + m i − n i and for each l ≥ the function Q l ( w ) is a polynomial of degree l whose coefficients do not dependon k .Proof. Consider first the function g k ( z ; α, β ) = ( βz ; q ) k ( αz ; q ) k . We compute g k ( z ; α, β ) = k − Y j =0 − βzq j − αzq j = ( β/α ) k k − Y j =0 − βz q − j − αz q − j and ln( g k ( z ; α, β )) = k ln( β/α ) + k − X j =0 (cid:16) ln(1 − β − z − q − j ) − ln(1 − α − z − q − j ) (cid:17) = k ln( β/α ) − k − X j =0 X l ≥ lz l q − jl (cid:16) β − l − α − l (cid:17) = k ln( β/α ) − X l ≥ lz l × (cid:16) β − l − α − l (cid:17) k − X j =0 q − jl = k ln( β/α ) − X l ≥ lz l × (cid:16) β − l − α − l (cid:17) − q − kl − q − l . Here in the second step we used Taylor series for ln(1 − w ) . The above series converges for all z large enough. We can express this result in the following form: for z large enough ln( g k ( z ; α, β )) = k ln( β/α ) + X l ≥ z − l P l ( q − k ) S.I. KALMYKOV, D. KARP, AND A. KUZNETSOV for some polynomials P l of degree l . After exponentiating both sides of the above identity andcollecting the powers of z − l on the right-hand side we obtain (for all z large enough)(14) g k ( z ; α, β ) = ( β/α ) k X l ≥ z − l ˜ P l ( q − k ) where every function ˜ P l is a polynomial of degree l .Next we use the identity ( w ; q ) k + l = ( w ; q ) l ( wq l ; q ) k and express f k in the following form(15) f k ( z ) = − ( zq − b ; q ) k + m ( zq − a ; q ) k + n +1 = R ( z ) r Y i =1 g k ( z ; q − a i + n i +1 , q − b i + m i +1 ) , where R ( z ) = − ( zq − b ; q ) m ( zq − a ; q ) n +1 . The function R has series expansion (for all z large enough)(16) R ( z ) = z M − N − r X l ≥ c l z − l , for some coefficients c l . Using (14) we find the expansion(17) r Y i =1 g k ( z ; q − a i + n i +1 , q − b i + m i +1 ) = A k X l ≥ z − l ˜ Q l ( q − k ) , for some polynomials ˜ Q l of degree l . Combining (16) and (17) we arrive at the desired result(13). (cid:3) The above Lemma implies that the coefficient C − ( k ) at z − equals if M − N − r < − ; andequals A k Q p ( q − k ) if M − N − r = p − , p ∈ N , where Q p ( y ) is a polynomial of degree p in y .Hence, we will have, assuming Q − ( y ) ≡ , S ( z ) = − m min − X k = − n max α k z k + ∞ X k = − m min z k r X i =1 k + n i X j =0 γ k + n i i,j = − m min − X k = − n max α k z k + ∞ X k = − m min ( Az ) k Q p ( q − k )= − m min − X k = − n max α k z k + ∞ X k = − m min ( Az ) k ( α p, + α p, q − k + · · · + α p,p q − pk )= − m min − X k = − n max α k z k + ( Az ) − m min ∞ X j =0 ( Az ) j ( α p, + α p, q − j + m min + · · · + α p,p q − p ( j − m min ) )= − m min − X k = − n max α k z k + ( Az ) − m min (cid:18) α p, − Az + α p, q m min − Az/q + · · · + α p,p q pm min − Az/q p (cid:19) . Noting that the common denominator on the RHS equals ( Az ; 1 /q ) p +1 = ( W z, q ) p +1 by thedefinition of p we arrive at (8). (cid:3) Remark.
In a recent paper [14] Yamaguchi, among other things, established an identity for φ basic hypergeometric function [14, Lemma 3]. After renaming parameters and rearranging NEW IDENTITY FOR THE SUM OF PRODUCTS OF GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS7 the terms his formula can be written as follows: X i =1 c i ( q − b + a i ; q ) m − n i ( q − b + a i ; q ) m − n i z − n i ( q a i − a i ′ ; q ) n i ′ − n i +1 2 φ (cid:18) q b − a i , q b − a i q a i ′ − a i z (cid:19) (18) × φ (cid:18) q − b + a i + m − n i , q − b + a i + m − n i q − a i ′ + a i + n i ′ − n i W z (cid:19) = z − max( n ,n ) P l ( z ) , where i ′ = 3 − i , a − a / ∈ Z and m , m , n , n are integers satisfying m + m ≤ n + n , P l isa polynomial of a fixed degree l expressible in terms of m i , n i , and W = q P k =1 ( b k − a k + n k − m k ) − , c i = q a i q P k =1 ( b k − a i )( m k − n i ) q − P k =1 ( m k − n i )( m k − n i +1) / . After applying Heine’s transformation formula φ (cid:18) q a , q b q c z (cid:19) = ( q a + b − c z ; q ) ∞ ( z ; q ) ∞ φ (cid:18) q c − a , q c − b q c q a + b − c z (cid:19) to each φ function in the left-hand side of (18), renaming the parameters a j
7→ − a j , b j − b j , m j
7→ − m j and n j
7→ − n j and simplifying the result one can reduce (18) to the special case of(8).In the following theorem we give an analogue of Theorem 1 for the function t φ s with t ≤ s .The result is given in terms of the mixture of t φ s and t ˆ φ s functions. Using formula (7) it can berewritten in terms of the standard t φ s functions only. Theorem 2.
Assume that q is a complex number satisfying | q | < , t and r are integers suchthat ≤ t < r , a ∈ C r satisfies a i − a j / ∈ Z for ≤ i < j ≤ r and b ∈ C t is arbitrary; assumefurther that m ∈ Z t , n ∈ Z r . Then, for some complex numbers β j , the following identity holds (19) r X i =1 q a i ( q − b + a i ; q ) m − n i z − n i ( q a i − a [ i ] ; q ) n [ i ] − n i +1 t φ r − (cid:18) q b − a i q a [ i ] − a i W q ( t − r ) a i z (cid:19) × t ˆ φ r − (cid:18) q − b + a i + m − n i q − a [ i ] + a i + n [ i ] − n i z (cid:19) = max( − m min − ,p ′ ) X j = − n max β j z j , where m min = min ≤ i ≤ t ( m i ) , n max = max ≤ i ≤ r ( n i ) , M = t P i =1 m i , N = r P i =1 n i , p ′ = [( M − N − r + 1) / ( r − t )] and W = q r − r Y i =1 q a i t Y i =1 q − b i .Proof. Repeating the proof of Theorem 1 we compute S ( z ) = r X i =1 q a i ( q − b + a i ; q ) m − n j z − n i ( q a i − a [ i ] ; q ) n [ i ] − n i +1 t φ r − (cid:18) q b − a i q a [ i ] − a i W q ( t − r ) a i z (cid:19) t ˆ φ r − (cid:18) q − b + a i + m − n i q − a [ i ] + a i + n [ i ] − n i z (cid:19) = r X i =1 ∞ X k =0 z k − n i k X j =0 q a i ( q − b + a i ; q ) m − n i ( q b − a i ; q ) j ( q − b + a i + m − n i ; q ) k − j ( q a q − b q r − q ( t − r ) a i ) j ( q a i − a [ i ] ; q ) n [ i ] − n i +1 ( q a [ i ] − a i ; q ) j ( q − a [ i ] + a i + n [ i ] − n i ; q ) k − j ( q ; q ) j ( q ; q ) k − j × h ( − j q j ( j − / i r − t = r X i =1 ∞ X k =0 z k − n i k X j =0 γ ki,j = r X i =1 ∞ X k i = − n i z k i k i + n i X j =0 γ k i + n i i,j , S.I. KALMYKOV, D. KARP, AND A. KUZNETSOV where(20) γ ki,j = q a i ( q − b + a i ; q ) m − n i ( q b − a i ; q ) j ( q − b + a i + m − n i ; q ) k − j ( q a q − b q r − q ( t − r ) a i ) j ( q a i − a [ i ] ; q ) n [ i ] − n i +1 ( q a [ i ] − a i ; q ) j ( q − a [ i ] + a i + n [ i ] − n i ; q ) k − j ( q ; q ) j ( q ; q ) k − j h ( − j q j ( j − / i r − t . After a calculation similar to that in the proof of Theorem 1, we arrive at γ ki,j = ( − ( t − r +1) j q ( t − r +1) j ( j − / q a i ( q − b + a i − j ; q ) m + k − n i ( q a i − a [ i ] − j ; q ) n [ i ] + k − n i +1 ( q ; q ) j ( q ; q ) k − j h ( − j q j ( j − / i r − t = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k − n i ( q a i − a [ i ] − j ; q ) n [ i ] + k − n i +1 ( q ; q ) j ( q ; q ) k − j or(21) γ k + n i i,j = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k ( q a i − a [ i ] − j ; q ) n [ i ] + k +1 ( q ; q ) j ( q ; q ) k + n i − j . Again, according to (2) we have ( q ; q ) j = ∞ for j < so that also ( q ; q ) k − j = ∞ for j > k and thus definition (20) implies that γ k + n i i,j = 0 for j < and j > k + n i . This relations extenddefinition of γ k + n i i,j to arbitrary j , and in view of this convention, we have S ( z ) = ∞ X k = − n max z k r X i =1 k + n i X j =0 γ k + n i i,j . Next, note that k + m i ≥ for all i = 1 , . . . , r if k ≥ − m min . If − m min ≤ − n max , then k ≥ − m min for all terms in the above sum. Otherwise, if − m min > − n max we can write S ( z ) = − m min − X k = − n max α k z k + ∞ X k = − m min z k r X i =1 k + n i X j =0 γ k + n i i,j = − m min − X k = − n max α k z k + S ( z ) , where α k = r X i =1 k + n i X j =0 γ k + n i i,j . Next, for each k ∈ Z define the function f k ( z ) = − ( zq − b ; q ) k + m ( zq − a ; q ) k + n +1 . These functions are rational for all k ∈ Z . Further, if k ≥ − m min the expression in the numeratoris a polynomial and all poles come from the zeros of the denominator. If k + n i + 1 > for all i = 1 , . . . , r , then the poles of f k ( z ) are at the points: ( zq − a ; q ) k + n +1 = 0 ⇔ z = q a i − j , i = 1 , . . . , r and j = 0 , . . . , k + n i . Since the poles are simple, we have after simple calculation res z = q ai − j f k ( z ) = ( − j q j ( j − / q a i ( q − b + a i − j ; q ) m + k ( q a i − a [ i ] − j ; q ) n [ i ] + k +1 ( q ; q ) j ( q ; q ) k + n i − j = γ k + n i i,j . It is clear that f k ( z ) = M − N − r − ( r − t ) k X j = −∞ C j ( k ) z j . NEW IDENTITY FOR THE SUM OF PRODUCTS OF GENERALIZED BASIC HYPERGEOMETRIC FUNCTIONS9
It implies that C − ( k ) = 0 for M − N − r − ( r − t ) k < − , i.e. for ( r − t ) k > M − N − r + 1 .Then S ( z ) = ∞ X k = − m min C − ( k ) z k = P ( r − t ) k ≤ M − N − r +1 k = − m min C − ( k ) z k , − p ′ ≤ m min , − p ′ > m min Finally, S ( z ) = − m min − X − n max α k z k + S ( z ) = max( − m min − ,p ′ ) X j = − n max β j z j . (cid:3) References
1. Mark J. Ablowitz and Athanassios S. Fokas,
Complex variables: introduction and applications , second ed.,Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2003. MR 19890492. W. N. Bailey,
Generalized hypergeometric series , Cambridge Tracts in Mathematics and Mathematical Physics,No. 32, Stechert-Hafner, Inc., New York, 1964. MR 01851553. Jouhet F. Beukers F.,
Duality relations for hypergeometric series , B.London.Math.Soc. (2015), no. 2,343–358.4. A. Ebisu, Three-term relations for the hypergeometric series , Funkcialaj Ekvacioj (2012), no. 2, 255–283.MR 30125775. Thomas Ernst, A comprehensive treatment of q-calculus , Birkhäuser, 2012.6. Runhuan Feng, Alexey Kuznetsov, and Fenghao Yang,
A short proof of duality relations for hypergeometricfunctions , J. Math. Anal. Appl. (2016), no. 1, 116–122. MR 35084827. George Gasper and Mizan Rahman,
Basic hypergeometric series , second ed., Encyclopedia of Mathematicsand its Applications, vol. 96, Cambridge University Press, Cambridge, 2004, With a foreword by RichardAskey. MR 21287198. Victor Kac and Pokman Cheung,
Quantum calculus , Universitext, Springer-Verlag, New York, 2002.MR 18657779. S. I. Kalmykov and D. B. Karp,
New identities for a sum of products of the Kummer functions , Sib. Èlektron.Mat. Izv. (2018), 267–276. MR 378335010. D.B. Karp and Kuznetsov A., A new identity for a sum of products of the generalized hypergeometric functions ,AMS Proc. (2020).11. Editors: M.Zuhair Nashed and Xin Li,
Frontiers in orthogonal polynomials and q-series , first ed., Contempo-rary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes, vol. 1, World Scientific,2018.12. Lucy Joan Slater,
Generalized hypergeometric functions , Cambridge University Press, Cambridge, 1966.MR 020168813. Hiroyuki Tagawa Victor J. W. Guo, Masao Ishikawa and Jiang Zeng,
A quadratic formula for basic hyperge-ometric series related to askey-wilson polynomials , AMS Proc. (2015), no. 4. MR 331411014. Yuka Yamaguchi,
Three-term relations for basic hypergeometric series , J. Math. Anal. Appl. (2018), no. 1,662–678. MR 3794109
School of mathematical sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai200240, China
E-mail address : [email protected] Faculty of Mathematics and Statistics, Ton Duc Thang University, 19 Ngyuen Huu Tho Street,Ho Chi Minh City, Vietnam
E-mail address , Corresponding author: [email protected]
Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3Canada
E-mail address ::