Generalizing the Index of the Deformed Rogers-Szegö Polynomials and the $q$-Exponential Operator

Abstract: This paper introduces the deformed Rogers-Szeg\"o functions ${\rm R}{\alpha}(x,y;u,v|q)$. When $\alpha=-n$ is a negative integer, these functions are related to the $q$-derivatives of Ramanujan's partial Theta function. Basic properties of the polynomial ${\rm R}{\alpha}$ are given, along with recurrence relations, its representations, and generating functions. We use the $u$-deformed $q$-exponential operator ${\rm T}(qD_{q}|u)$ to obtain identities for Rogers-Szeg\"o functions, in particular, Rogers-type formulas.