Recurrence coefficients for discrete orthonormal polynomials and the Painlev\'e equations

TL;DR

This paper explores how recurrence coefficients of certain discrete orthogonal polynomials relate to special function solutions of Painlevé equations, revealing new connections between orthogonal polynomials and integrable systems.

Contribution

It demonstrates that recurrence coefficients for semi-classical Charlier and Meixner polynomials can be expressed via Wronskians of special functions linked to Painlevé equations, a novel connection.

Findings

Recurrence coefficients are expressed in terms of Wronskians of special functions.

Connections established between discrete orthogonal polynomials and Painlevé equations.

Provides explicit formulas linking orthogonal polynomial coefficients to integrable systems.

Abstract

We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painlev\'e equations.