A generating function and formulae defining the first-associated Meixner-Pollaczek polynomials

TL;DR

This paper introduces a new class of first-associated Meixner-Pollaczek polynomials derived from shift operators with a specific sequence, providing their generating function and a related integral transform with a Humbert's function kernel.

Contribution

It defines the first-associated Meixner-Pollaczek polynomials via a generating function and constructs a Bargmann-type integral transform with a Humbert's function kernel.

Findings

Derived the generating function for the polynomials.

Constructed a Bargmann-type integral transform.

Connected the polynomials to nonlinear coherent states.

Abstract

While considering nonlinear coherent states with specific anti-holomorphic coefficients $\bar{z}^n/\sqrt{x_n!}$, we identify as first associated Meixner-Pollaczek polynomials the orthogonal polynomials arising from shift operators which are defined by the sequence $x_n=(n+1)^2$ . We give a formula defining these polynomials by writing down their generating function. This also leads to construct a Bargmann-type integral transform whose kernel is given in terms of a $\Psi_1$ Humbert's function.