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.

We have added several new terminating balanced, 2-balanced and 3-balanced {}_4\phi_3 summations. Corresponding to the balanced {}_4\phi_3 summations we have new quadratic special values for the Askey--Wilson polynomials and have computed corresponding new transformations for products of two nonterminating {}_2\phi_1 basic hypergeometric functions