Generating functions for some families of the generalized Al-Salam–Carlitz q-polynomials

Abstract

Abstract In this paper, by making use of the familiar q-difference operators D q $D_{q}$ and D q − 1 $D_{q^{-1}}$ , we first introduce two homogeneous q-difference operators T ( a , b , c D q ) $\mathbb{T}(\mathbf{a},\mathbf{b},cD_{q})$ and E ( a , b , c D q − 1 ) $\mathbb{E}(\mathbf{a},\mathbf{b}, cD_{q^{-1}})$ , which turn out to be suitable for dealing with the families of the generalized Al-Salam–Carlitz q-polynomials ϕ n ( a , b ) ( x , y | q ) $\phi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)$ and ψ n ( a , b ) ( x , y | q ) $\psi_{n}^{(\mathbf{a},\mathbf{b})}(x,y|q)$ . 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.

Keywords

• Basic (or q-) hypergeometric series
• Homogeneous q-difference operator
• q-Binomial theorem
• Cauchy polynomials
• Al-Salam–Carlitz q-polynomials
• Rogers type formulas