Recurrence Coefficients of a New Generalization of the Meixner Polynomials

TL;DR

This paper introduces new generalizations of Meixner polynomials on various lattices, linking their recurrence coefficients to solutions of the fifth Painlevé equation, and explores related Bäcklund transformations.

Contribution

It establishes a connection between generalized Meixner polynomial recurrence coefficients and solutions of P_V, including initial conditions and properties of Bäcklund transformations.

Findings

Recurrence coefficients relate to P_V solutions.

Initial conditions correspond to classical P_V solutions.

Analyzes Bäcklund transformation properties.

Abstract

We investigate new generalizations of the Meixner polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term recurrence relation for the orthogonal polynomials are related to the solutions of the fifth Painlev\'e equation P$_{\textup V}$. Initial conditions for different lattices can be transformed to the classical solutions of P$_{\textup V}$ with special values of the parameters. We also study one property of the B\"acklund transformation of P$_{\textup V}$.