Discrete Orthogonal Polynomials with Hypergeometric Weights and Painlevé VI

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 Painlevé equations, and the differential equation is the σ-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as n→∞ using the discrete Painlevé equations.