Generalized deformed oscillators in framework of unified (q;\alpha,\beta,\gamma;\nu)-deformation and their oscillator algebras

TL;DR

This paper reviews multi-parameter deformed oscillators, introduces a generalized algebra framework, explores their representations, links to special polynomials, and constructs associated coherent states with proven completeness.

Contribution

It introduces a new generalized (q;α,β,γ;ν)-deformed oscillator algebra and analyzes its properties and representations, extending existing oscillator models.

Findings

Connection of the generalized oscillator with Askey (1/q)-Hermite polynomials

Construction of generalized coherent states with explicit formulas

Proof of overcompleteness of the coherent states

Abstract

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;\alpha,\beta,\gamma;\nu)-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.