A q deformation of true-polyanalytic Bargmann transforms when q^{-1}> 1

TL;DR

This paper introduces a new q-deformation of the true-polyanalytic Bargmann transform using orthogonal polynomials, with potential applications in q-deformed point processes and quantum oscillators.

Contribution

It constructs a novel q-deformed Bargmann transform based on combined orthogonal polynomials, extending known results to a new q-analogue setting.

Findings

Recovered known Arik-Coon oscillator result for m=0

Developed a new q-deformation of the polyanalytic Bargmann transform

Potential application to q-deformed Ginibre point processes

Abstract

We combine continuous $q^{-1}$-Hermite Askey polynomials with new $2D$ orthogonal polynomials introduced by Ismail and Zhang as $q$-analogs for complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter $m$. In the analytic case corresponding to $m=0$, we recover a known result on the Ar\"{\i}k-Coon oscillator for $q'=q^{-1}>1$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform on the complex plane. The obtained result may be used to introduce a $q$-deformed Ginibre-type point process.