A Note on the Rogers-Szegö Polynomial $q$-Differential Operators

Abstract

In this paper, we introduce the Rogers-Szegö deformed $q$-differential operators g${}_n(b{D}_q|u)$ based on $q$-differential operator $D_q$. The motivation for introducing the operators g${}_n(b{D}_q)$ is that their limit turns out to be the $q$-exponential operator T$(b{D}_q)$ given by Chen. The deformed homogeneous Al-Salam-Carlitz polynomials ${\Psi }_m^{(q^{-n})}(ub,x|u{q}^{-1})$ can easily be represented by using the operators g${}_n(b{D}_q|u)$. Identities relating the new general Al-Salam-Carlitz polynomial, defined by Cao et al., the generalized, and homogeneous Al-Salam-Carlitz polynomials ${\Phi }_m^{(q^n)}(b,x|q)$ and basic hypergeometric series are given.