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 two new terminating balanced ${}_4\phi_3$ summations that give rise to quadratic special values for Askey--Wilson polynomials. We also present three new terminating 2-balanced ${}_4\phi_3$ summations and two new terminating 3-balanced ${}_4\phi_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $_4\phi_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations.