Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials

Abstract

Ismail and Wilson derived a generating function for Askey--Wilson polynomials which is given by a product of $q$-Gauss (Heine) nonterminating basic hypergeometric functions. We provide a generalization of that generating function which contains an extra parameter. A special case gives a closed form summation formula for a quadruple basic hypergeometric sum. We further present new terminating balanced ${}_4\ddot{I}{•}_3$ summations that give rise to $q$-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced ${}_4\ddot{I}{•}_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric ${}_4\ddot{I}{•}_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations. Further new identities are obtained by applying Cayley--Orr type expansion formulas.

The following changes were made: (1) extra references to the literature was added to give further context (2) a missing $q$-quadratic product transformation Theorem 4.3 was added along with its $q\to 1^{-}$ limits (70), (71)