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The bivariate Rogers-Szegö polynomials. (English) Zbl 1119.05011
Two classical results for the Rogers-Szegő polynomials are Mehler’s formula and Roger’s formula which respectively correspond to the Poisson kernel formula and the linearization formula. In this paper, the authors extend these formulae to the bivariate Rogers-Szegő polynomials $h_n(x,y \vert q)$ by using the $q$-exponential operator as studied by {\it W. Y. C. Chen} and {\it Z. Liu} [J. Comb. Theory, Ser. A 80, No. 2, 175--195 (1997; Zbl 0901.33009)] and the homogeneous $q$-shift operator introduced by {\it W. Y. C. Chen, A. M. Fu} and {\it B. Zhang} [Adv. Appl. Math. 31, No. 4, 659--668 (2003; Zbl 1075.39018)]. This new approach to the nonsymmetric Poisson kernel for the continuous big $q$-Hermite polynomials is easier to deal with from the operator point of view. These results can be used to compute some integrals with the aid of the Askey-Wilson integral.

MSC:
05A30$q$-calculus and related topics
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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