Two expansion formulas involving the Rogers–Szegő polynomials with applications

Abstract

Using two expansion formulas for the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials, we give new proofs of a variety of important classical formulas including Bailey's 6ψ6 summation, the Askey–Wilson integral and its extension. Furthermore, we give nontrivial extensions of the Andrews multiple version of the Rogers–Selberg identity, as well as the Sylvester identity.