Characterization of $({\cal R},p,q)-$deformed Rogers-Szeg\"o polynomials: associated quantum algebras, deformed Hermite polynomials and relevant properties

TL;DR

This paper introduces a unified framework for $({\cal R},p,q)$-deformed Rogers-Szeg"o polynomials, deriving their recurrence relations, associated quantum algebras, and deformed Hermite polynomials, generalizing previous results.

Contribution

It provides a new, comprehensive characterization of $({\cal R},p,q)$-deformed polynomials and their algebraic structures, unifying and extending known results.

Findings

Derived three-term recurrence relations for the polynomials

Constructed associated quantum algebras with creation and annihilation operators

Introduced and analyzed deformed Hermite polynomials

Abstract

This paper addresses a new characterization of $({\cal R},p,q)-$deformed Rogers-Szeg\"o polynomials by providing their three-term recurrence relation and the associated quantum algebra built with corresponding creation and annihilation operators. The whole construction is performed in a unified way, generalizing all known relevant results which are straightforwardly derived as particular cases. Continuous $({\cal R},p,q)-$deformed Hermite polynomials and their recurrence relation are also deduced. Novel relations are provided and discussed.