Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials
Howard Cohl, Michael Schlosser
TL;DR
The paper advances the theory of basic hypergeometric series by generalizing the Ismail–Wilson generating function for Askey–Wilson polynomials to include an extra parameter, yielding a triple-sum expansion and a closed-form quadruple sum. It then develops and catalogs new terminating ${}_4 extphi{}_3$ summations, including quadratic, 2-balanced, and 3-balanced cases, and derives quadratic special values for $p_n(0; , |q)$. By coupling these terminating sums with the generalized generating function, the authors obtain nonterminating product transformations and their integral representations, thereby expanding the toolbox for manipulating products of nonterminating basic hypergeometric series. They also outline alternative verification routes via Cayley–Orr type identities and Gasper–Rahman/Andrews frameworks, underscoring the robustness of the results and their potential applications in q-series and representation theory.
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 ${}_4φ_3$ summations that give rise to quadratic special values for Askey--Wilson polynomials. We also present three new terminating 2-balanced ${}_4φ_3$ summations and two new terminating 3-balanced ${}_4φ_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $_4φ_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations.