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$(q{D}_q|u)$, when $u=q^b,q^{b+1/2}$, $b\ge 1$, are obtained. This allows us to obtain some $q^b$-series and $q^{b+1/2}$-series. Additionally, transformation formulas for basic hypergeometric series ${}_1{\varphi }_1$ and ${}_1{\varphi }_2$ are derived.