A Note on the Rogers-Szeg\"{o} Polynomial $q$-Differential Operators

In this paper, we introduce the Rogers-Szeg\"o deformed $q$-differential operators g$_{n}(bD_{q}|u)$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators g$_{n}(bD_{q})$ is that their limit turns out to be the $q$-exponential operator T$(bD_{q})$ given by Chen. The deformed homogeneous Al-Salam-Carlitz polynomials $\Psi_{m}^{(q^{-n})}(ub,x|uq^{-1})$ can easily be represented by using the operators g$_{n}(bD_{q}|u)$. 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.