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

In this paper, we introduce the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szeg\"o polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szeg\"o 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 $\mathrm{E}(yD_{q}|u)$, defined here. From this operator, we obtain for free the operators T$(yD_{q})$ the Chen, $\mathrm{R}(yD_{q})$ of Saad, $\mathcal{E}(yD_{q})$ of Exton, and $\mathcal{R}(yD_{q})$ of Rogers-Ramanujan when $u=1,q,\sqrt{q},q^2$, respectively. We introduce the deformed basic hypergeometric series ${}_{r}\Phi_{s}$, a generalization of the classical basic hypergeometric series. New transformation formulas for basic hypergeometric series are obtained.