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 4φ3 summations that give rise to q-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced 4φ3 summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric 4φ3 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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