Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlev\'e VI

TL;DR

This paper studies discrete orthogonal polynomials with hypergeometric weights, revealing their recurrence coefficients satisfy Painlevé equations and analyzing their asymptotic behavior as the degree grows large.

Contribution

It establishes a connection between recurrence coefficients of these polynomials and Painlevé equations, providing new insights into their structure and asymptotics.

Findings

Recurrence coefficients satisfy discrete Painlevé equations.

The differential equation is the σ-form of Painlevé VI.

Asymptotic analysis of coefficients as n approaches infinity.

Abstract

We investigate the recurrence coefficients of discrete orthogonal polynomials on the non-negative integers with hypergeometric weights and show that they satisfy a system of non-linear difference equations and a non-linear second order differential equation in one of the parameters of the weights. The non-linear difference equations form a pair of discrete Painlev\'e equations and the differential equation is the $\sigma$-form of the sixth Painlev\'e equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlev\'e equations.