$q$-deformed coherent states associated with the sequence $x_n^{q,\alpha }=(1+\alpha q^{n-1})[n]_q$

TL;DR

This paper introduces a new class of $q$-deformed coherent states using a generalized factorial sequence, providing a unified framework that interpolates between different known $q$-coherent states and exploring their mathematical properties.

Contribution

The authors define generalized $q$-coherent states based on a new factorial sequence and realize them via Al-Salam-Chihara polynomials, extending the theory of $q$-coherent states.

Findings

States interpolate between Arik-Coon and Barut-Girardello types.

Realization in terms of Al-Salam-Chihara polynomials.

Discussion of associated Bargmann transforms.

Abstract

We introduce new generalized $q$-deformed coherent states ($q$-CS) by replacing the $q$-factorial of $[n]_q!$ in the series expansion of the classical $q$-CS by the generalized factorial $x_n^{q,\alpha}!$ where $x_n^{q,\alpha}=(1+\alpha q^{n-1})[n]_q$. We use the shifted operators method based on the sequence $x_n^{q,\alpha}$ to obtain a realization in terms of Al-Salam-Chihara polynomials for the basis vectors of the Fock space carrying the constructed $q$-CS. These new states interpolate between the $q$-CS of Arik-Coon type ($\alpha=0$, $0<q<1$) and a set of coherent states of Barut-Girardello type for the Meixner-Pollaczek oscillator ($\alpha\neq 0$, $q\to 1$). We also discus their associated Bargmann type transforms.