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

We introduce new generalized q-deformed coherent states (q-CS) by replacing the q-factorial of \([n]_q!\) ${[n]}_q!$ in the series expansion of the classical q-CS by the generalized factorial \(x_n^{q,\alpha }!\) $x_n^{q,\alpha }!$ where \(x_n^{q,\alpha }=(1+\alpha q^{n-1})[n]_q\) $x_n^{q,\alpha }=(1+\alpha q^{n-1}){[n]}_q$ . We use the shifted operators method based on the sequence \(x_n^{q,\alpha }\) $x_n^{q,\alpha }$ 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 ( \(\alpha =0\) $\alpha =0$ , \(0<q<1\) $0<q<1$ ) and a set of coherent states of Barut–Girardello type for the Meixner–Pollaczek oscillator ( \(\alpha \ne 0\) $\alpha \ne 0$ , \(q\rightarrow 1\) $q\to 1$ ). We also discuss their associated Bargmann-type transforms. As application, we introduce a generalization of the Euler probability distribution and we derive its main statistical parameters.