A q-operational equation and the Rogers-Szegő polynomials

By solving a q-operational equation with formal power series, we prove a new q-exponential operational identity. This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő polynomials. With this operational identity, we can easily derive, among others, the q-Mehler formula, the q-Burchnall formula, the q-Nielsen formula, the q-Doetsch formula, the q-Weisner formula, and the Carlitz formula for the Rogers-Szegő polynomials. This operational identity also provides a new viewpoint on some other basic q-formulas. It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.