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The Bivariate Rogers-Szeg\"{o} Polynomials
We present an operator approach to deriving Mehler's formula and the Rogers
formula for the bivariate Rogers-Szeg\"{o} polynomials . The proof
of Mehler's formula can be considered as a new approach to the nonsymmetric
Poisson kernel formula for the continuous big -Hermite polynomials
due to Askey, Rahman and Suslov. Mehler's formula for
involves a sum and the Rogers formula involves a sum.
The proofs of these results are based on parameter augmentation with respect to
the -exponential operator and the homogeneous -shift operator in two
variables. By extending recent results on the Rogers-Szeg\"{o} polynomials
due to Hou, Lascoux and Mu, we obtain another Rogers-type formula
for . Finally, we give a change of base formula for
which can be used to evaluate some integrals by using the Askey-Wilson
integral.Comment: 16 pages, revised version, to appear in J. Phys. A: Math. Theo