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 φ 2 sum and the Rogers formula involves a 2 φ 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.Keywords: The bivariate Rogers-Szegö polynomials, the q-exponential operator, the homogeneous q-shift operator, Mehler's formula, the Rogers formula, Askey-Wilson integral.
We define a q-exponential operator R(bD q) which turn out to be suitable for dealing with the Cauchy polynomials P n (x, y) and the homogeneous Rogers-Szegö polynomials h n (x, y|q). By using this operator, we derive Mehler's formula and Rogers formula for the polynomials P n (x, y) and h n (x, y|q).
Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients. Our approach easily extends to the q-analogues.
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