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 $\sigma$-form of the sixth Painlevé equation. We briefly investigate the asymptotic behavior of the recurrence coefficients as $n\to \infty$ using the discrete Painlevé equations.

References in corpus (2)

• Recurrence Coefficients of a New Generalization of the Meixner Polynomials
• The generalized Krawtchouk polynomials and the fifth Painlevé equation

Cited by in corpus (3)

• Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations
• Asymptotic behaviours of q-orthogonal polynomials from a q-Riemann Hilbert Problem
• Discrete multiple orthogonal polynomials on shifted lattices