A generating function for non-standard orthogonal polynomials involving differences: the Meixner case

TL;DR

This paper derives a generating function for non-standard Meixner orthogonal polynomials involving difference-based inner products, and extends the results to Laguerre polynomials via a limit process.

Contribution

It introduces a generating function for Δ-Meixner–Sobolev orthogonal polynomials and connects it to Laguerre–Sobolev polynomials through a limiting procedure.

Findings

Derived a generating function for Δ-Meixner–Sobolev polynomials.

Established a limit process to recover Laguerre–Sobolev generating functions.

Enhanced understanding of orthogonal polynomials with difference-based inner products.

Abstract

In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $\Delta$--Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the $\Delta$--Meixner--Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre--Sobolev orthogonal polynomials.