Deformed Homogeneous Polynomials and the Generalized $q$-Exponential Operator

Abstract

In this paper, we introduce the deformed homogeneous polynomials ${\mathrm{R}}_n(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szegö polynomials ${\mathrm{h}}_n(x|q)$, the generalized Rogers-Szegö polynomials ${\mathrm{r}}_n(x,y)$, the Stieltjes-Wigert polynomials ${\mathrm{S}}_n(x;q)$, among others. Basic properties of the polynomial ${\mathrm{R}}_n$ are given, along with recurrence relations, its $q$-difference equation, and representations. Generating functions for the polynomials ${\mathrm{R}}_n(x,y;u|q)$ are given. These functions include generalizations of the Mehler and Rogers formulas. In addition, generalizations of the $q$-binomial formula and the Heine transformation formula are obtained. These results are obtained via the $u$-deformed $q$-exponential operator $\mathrm{E}(y{D}_q|u)$, defined here. From this operator, we obtain for free the operators T$(y{D}_q)$ the Chen, $\mathrm{R}(y{D}_q)$ of Saad, $\mathcal{E}(y{D}_q)$ of Exton, and $\mathcal{R}(y{D}_q)$ of Rogers-Ramanujan when $u=1,q,\sqrt{q},q^2$, respectively. We introduce the deformed basic hypergeometric series ${}_r\hat{I}{¦}_s$, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.