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arXiv · 2411.03571

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

Howard Cohl·Michael Schlosser
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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 ${}_4\phi_3$ summations that give rise to $q$-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced ${}_4\phi_3$ 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. Further new identities are obtained by applying Cayley--Orr type expansion formulas.

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Howard Cohl, Michael Schlosser. 2024-11-06. Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials. https://arxiv.org/abs/2411.03571

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