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

El Moize, Othmane; Mouayn, Zouhaïr

Comptes Rendus. Mathématique, Tome 359 (2021) no. 10, pp. 1295-1305.

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.

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