$(q,\mu)$ and $(p,q,\zeta)-$exponential functions: Rogers-Szeg\H{o} polynomials and Fourier-Gauss transform

TL;DR

This paper explores generalized exponential functions related to $(q,u)$ and $(p,q,u)$-oscillator algebras, deriving generating functions, matrix elements, and Fourier-Gauss transforms to advance understanding of deformed exponential functions and Rogers-Szeg
51 polynomials.

Contribution

It introduces new matrix element computations and generating functions for generalized deformed exponential functions linked to oscillator algebras, extending prior work on Rogers-Szeg
51 polynomials.

Findings

Derived matrix elements for $(q,u)$ and $(p,q,u)$-oscillator algebras.

Established generating functions for Rogers-Szeg
51 polynomials.

Performed Fourier-Gauss transform on generalized deformed exponential functions.

Abstract

From the realization of $q-$oscillator algebra in terms of generalized derivative, we compute the matrix elements from deformed exponential functions and deduce generating functions associated with Rogers-Szeg\H{o} polynomials as well as their relevant properties. We also compute the matrix elements associated to the $(p,q)-$oscillator algebra (a generalization of the $q-$one) and perform the Fourier-Gauss transform of a generalization of the deformed exponential functions.