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

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 $m$. Our construction leads to a new $q$-deformation of the $m$-true-polyanalytic Bargmann transform on the complex plane. In the analytic case $m=0$, the obtained coherent states transform can be associated with the Arïk-Coon oscillator for ${q^{\prime}=q^{-1}>1}$. These result may be used to introduce a $q$-deformed Ginibre-type point process.