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 $${\phantom {a}}_4\phi _3$$ a 4 ϕ 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 $${\phantom {a}}_4\phi _3$$ a 4 ϕ 3 summations. Using the Ismail–Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $${\phantom {a}}_4\phi _3$$ a 4 ϕ 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.