Homogeneous $q$-difference Equations and Generating Functions for the Generalized 2D-Hermite Polynomials

Abstract

In this paper, we deduce several types of generating functions for $q$-2D Hermite polynomial by the method of homogeneous $q$-difference equations. Besides, we deduce a multilinear generating function for $q$-2D Hermite polynomials as a generalization of Andrew's result. Moreover, we build a transformation identity involving the generalized $q$-2D Hermite polynomials by the method of homogeneous $q$-difference equations. As an application, we give a transformation identity involving $D_{q}(m,n)$ and $D_{q}^{*}(m,n)$.