The generalized Hahn polynomials

Abstract

In this paper, we represent the generalized Hahn polynomials ϕ(a)n (x, y) by the Cauchy operator for deriving its identities: generating function, Mehler’s formula, Rogers formula (with some of its applications), Rogers-type formula, extended generating function, extended Mehler’s formula, extended Rogers formula and another extended identities. Also, the Rogers-type formula for the bivariate (generalized) (classical) Rogers-Szeg¨o polynomials will be given by two methods. Then we give the q-integral representation for the generalized Hahn polynomials, bivariate Rogers-Szeg¨o polynomials, and the generalized Rogers-Szegö polynomials.