Generalizing the Index of the $u$-Deformed Homogeneous Polynomials and Generating Functions for $\mathrm{R}_{-n}(x,y;q|u)$

Abstract

This paper introduces the $u$-deformed homogeneous functions $\mathrm{R}_{\alpha}(x,y;u|q)$, for all $\alpha\in\mathbb{C}$. Basic properties of the functions $\mathrm{R}_{\alpha}(x,y;u|q)$ are given, along with recurrence relations, their $q$-difference equation, and representations. Generating functions for the functions $\mathrm{R}_{-n}(x,y;u|q)$ via the $u$-deformed $q$-exponential operator E$(qD_{q}|u)$, when $u=q^b,q^{b+1/2}$, $b\geq1$, are obtained. This allows us to obtain some $q^b$-series and $q^{b+1/2}$-series. Additionally, transformation formulas for basic hypergeometric series $_{1}\phi_{1}$ and $_{1}\phi_{2}$ are derived.