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

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 ($\alpha \ne 0$, $q\to 1$). We also discus their associated Bargmann type transforms.