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 $h_n(x,y|q)$. 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 ${}_3{\varphi }_2$ sum and the Rogers formula involves a ${}_2{\varphi }_1$ 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.

16 pages, revised version, to appear in J. Phys. A: Math. Theor

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• More on the q-oscillator algebra and q-orthogonal polynomials
• Orientation of particle attachment and local isotropy in diffusion limited aggregation (DLA)

Cited by in corpus (6)

• An Operator Approach to the Al-Salam-Carlitz Polynomials
• Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials
• Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions
• Multivariate Rogers-Szegö polynomials and flags in finite vector spaces
• On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications
• The Cauchy Operator for Basic Hypergeometric Series