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

TL;DR

The article generalizes Rogers-Szegö polynomials through the index-parametrized functions $R_{oldsymbol{\alpha}}(x,y;u,v|q)$ and develops a comprehensive toolkit around the $u$-deformed $q$-exponential operator. It establishes recurrence relations, negative-index representations via Ramanujan's partial Theta function, and generating functions expressed in terms of basic hypergeometric series. The work ties negative-index cases to $q$-derivatives of partial Theta functions and furnishes broad Rogers-type identities, showcasing the utility of the operator framework for deformed polynomials. This advances the understanding of deformed Rogers-Szegö structures and their connections to theta functions, with potential implications for $q$-series and combinatorial identities.

Abstract

This paper introduces the deformed Rogers-Szegö functions ${\rm R}_α(x,y;u,v|q)$. When $α=-n$ is a negative integer, these functions are related to the $q$-derivatives of Ramanujan's partial Theta function. Basic properties of the polynomial ${\rm R}_α$ 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-Szegö functions, in particular, Rogers-type formulas.

Theorems & Definitions (31)

• Lemma 1
• Proposition 1
• Definition 1
• Proposition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof