Generating Functions for Some Families of the Generalized Al-Salam-Carlitz $q$-Polynomials

Abstract

In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $\phi_n^{({\bf a},{\bf b})}(x,y|q)$ and $\psi_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivastava-Agarwal type linear as well as bilinear generating functions involving each of these families of the generalized Al-Salam-Carlitz $q$-polynomials. We also show how the various results presented here are related to those in many earlier works on the topics which we study in this paper.