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arXiv:2102.10714

A set of $q$-coherent states for the Rogers-Szegő oscillator

Othmane El Moize, Zouhaı̈r Mouayn

Published Feb 21, 2021• 0 citations
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Abstract

We discuss a model of a $q$-harmonic oscillator based on Rogers-Szegő 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.

DOI
10.48550/arxiv.2102.10714

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A set of qqq-coherent states for the Rogers-Szegő oscillator
We discuss a model of a qqq-harmonic oscillator based on Rogers-Szegő functions. We combine these functions with a class of qqq-analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegative integer parameter mmm. Our construction leads to a new qqq-deformation of the mmm-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 qqq-deformation of the Ginibre-mmm-type process on the complex plane.
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 qdeformation 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 = q -1 > 1.These result may be used to introduce a q-deformed Ginibre-type point process.
In this paper we study the Arik–Coon oscillator with the main relation aa + − qa + a = 1, where q > 1, in the framework of the unified ( q ; α, β, γ; ν)-deformed oscillator algebra. This oscillator is connected with the Askey q −1 -Hermite polynomials. We construct a family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical moment problem, we prove the over-completeness of this family of coherent states.
The aim of this paper is to review our results on description of the multi-parameter deformed oscillators and their oscillator algebras. We define generalized (q;α,β,γ;ν)-deformed oscillator algebra and study its irreducible representations. The Arik-Coon oscillator with the main relation aa^+ - q a^+a = 1, where q > 1 is embedded in this framework. We find connection of this oscillator with the Askey (1/q)-Hermite polynomials. We construct family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of appropriate classical Stielties moment problem we prove the (over)completeness relation of these states.
The aim of this paper is to review our results on the description of multiparameter deformed oscillators and their oscillator algebras. We define generalized (q; α, β, γ, ν)-deformed oscillator algebras and study their irreducible representations. The Arik–Coon oscillator with the main relation aa+ – qa+a = 1, where q >1, is embedded in this framework. We have found the connection of this oscillator with the Askey q–1-Hermite polynomials. We construct a family of generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical Stieltjes moment problem, we prove the property of (over)completeness of these states.