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

TL;DR

This paper develops the theory of $u$-deformed homogeneous functions, deriving their properties, recurrence relations, and generating functions, and applies these to obtain new series and transformation formulas for basic hypergeometric series.

Contribution

It introduces the $u$-deformed homogeneous functions $ ext{R}_ ext{alpha}$, explores their properties, and derives generating functions and transformation formulas, extending the understanding of these functions in $q$-series.

Findings

Derived recurrence relations and $q$-difference equations for $ ext{R}_ ext{alpha}$.

Obtained generating functions for $ ext{R}_{-n}$ using the $u$-deformed $q$-exponential operator.

Established transformation formulas for $_{1}oldsymbol{ ext{phi}}_{1}$ and $_{1}oldsymbol{ ext{phi}}_{2}$ series.

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.