Closed-form expressions, generating function and Mehler-Heine type formula for the associated Meixner polynomials and some applications

TL;DR

This paper derives explicit formulas and generating functions for associated Meixner, Charlier, and Laguerre polynomials, investigates their asymptotic behavior, and discusses applications of these results.

Contribution

It provides new closed-form expressions and generating functions for associated orthogonal polynomials, along with asymptotic formulas and applications.

Findings

Closed-form expressions for associated Meixner, Charlier, and Laguerre polynomials.

Derived generating functions for these polynomials.

Established Mehler-Heine type asymptotic formulas.

Abstract

We give a closed-form expression for the associated Meixner polynomials from which we derive closed-form expressions for the associated Charlier and Laguerre polynomials by a limit procedure. These formulas are then used to derive generating functions of these polynomials. Also, we investigate the Mehler-Heine type asymptotic formulas for the associated Meixner and Charlier polynomials. Some applications and consequences of our results are also mentioned.