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

TL;DR

This paper introduces new $q$-differential operators related to Rogers-Szeg"o polynomials, explores their limits, and demonstrates their use in representing and deriving identities for various Al-Salam-Carlitz polynomials and hypergeometric series.

Contribution

The paper presents novel $q$-differential operators g$_{n}(bD_{q}|u)$ and their applications to polynomial representations and identities, extending the theory of Al-Salam-Carlitz polynomials.

Findings

Defined new $q$-differential operators g$_{n}(bD_{q}|u)$.

Expressed deformed homogeneous Al-Salam-Carlitz polynomials using these operators.

Derived identities relating various Al-Salam-Carlitz polynomials and hypergeometric series.

Abstract

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.