Laguerre-Freud Equations for the Generalized Charlier, Generalized Meixner and Gauss Hypergeometric Orthogonal Polynomials

TL;DR

This paper derives Laguerre-Freud equations for generalized Charlier, Meixner, and Gauss hypergeometric orthogonal polynomials, revealing connections to Painlevé equations and comparing with existing results.

Contribution

It provides new derivations of Laguerre-Freud relations for these polynomials and explores their differential equations, extending and comparing prior findings.

Findings

Laguerre-Freud relations for generalized Charlier and Meixner polynomials are derived.

Differential equations for recursion coefficients are identified as Painlevé type.

Differences with previous equations for Gauss hypergeometric polynomials are analyzed.

Abstract

The Cholesky factorization of the moment matrix is considered for the generalized Charlier, generalized Meixner, and Gauss hypergeometric discrete orthogonal polynomials. For the generalized Charlier, we present an alternative derivation of the Laguerre-Freud relations found by Smet and Van Assche. Third-order and second-order nonlinear ordinary differential equations are found for the recursion coefficient $\gamma_n$, that happen to be forms of the Painlev\'e $\text{deg-P}_{\text V}$ in disguise. Laguerre-Freud relations are also found for the generalized Meixner case, which are compared with those of Smet and Van Assche. Finally, the Gauss hypergeometric discrete orthogonal polynomials, also known as generalized Hahn of type I, are also studied. Laguerre-Freud equations are found, and the differences with the equations found by Dominici and by Filipuk and Van Assche are provided.