The bivariate Rogers-Szegoe polynomials

We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegoe polynomials hn(x, y vertical bar 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 vertical bar q) due to Askey, Rahman and Suslov. Mehler's formula for hn(x, y vertical bar 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-Szegoe polynomials hn(x vertical bar q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for hn(x, y vertical bar q). Finally, we give a change of base formula for Hn(x; a vertical bar q) which can be used to evaluate some integrals by using the Askey-Wilson integral