The bivariate Rogers–Szegö polynomials

Abstract

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers–Szegö polynomials hn(x, y|q). The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials Hn(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y|q) involves a 3ϕ2 sum and the Rogers formula involves a 2ϕ1 sum. The proofs of these results are based on parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending recent results on the Rogers–Szegö polynomials hn(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y|q). Finally, we give a change of base formula for Hn(x; a|q) which can be used to evaluate some integrals by using the Askey–Wilson integral.