A set of $q$-coherent states for the Rogers-Szeg\H{o} oscillator

TL;DR

This paper introduces a novel set of $q$-coherent states for the Rogers-Szeg ext{"o} oscillator, combining $q$-functions and polynomials to generalize classical transforms and spaces with potential applications in $q$-deformed stochastic processes.

Contribution

It constructs new $q$-coherent states using Rogers-Szeg ext{"o} functions and $q$-Hermite polynomials, leading to a $q$-deformation of the polyanalytic Bargmann transform and related spaces.

Findings

Defined a new $q$-deformed Bargmann transform.

Derived an explicit reproducing kernel for the generalized space.

Proposed a $q$-deformation of the Ginibre-$m$-type process.

Abstract

We discuss a model of a $q$-harmonic oscillator based on Rogers-Szeg\H{o} functions. We combine these functions with a class of $q$-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform whose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernel of this space. The obtained results may be exploited to define a $q$-deformation of the Ginibre-$m$-type process on the complex plane.