The Bivariate Rogers-Szegö Polynomials
William Y.C. Chen, Husam L. Saad and Lisa H. Sun
Abstract: We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szegö polynomials h_{n}(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 H_{n}(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for h_{n}(x, y|q) involves a _{3}ø_{2} sum and the Rogers formula involves a _{2}ø_{1} 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 h_{n}(x|q) due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for h_{n}(x, y|q). Finally, we give a change of base formula for H_{n}(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral. AMS Classifications: 5A30, 33D45 Keywords: The bivariate Rogers-Szegö polynomials, the continuous big q-Hermite polynomials, the Cauchy polynomials, the q-exponential operator, the homogeneous q-shift operator Download: pdf |