Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlev\'e equations

TL;DR

This paper demonstrates how recurrence coefficients of discrete orthogonal polynomials with hypergeometric weights satisfy the discrete Painlevé-V equation, illustrating the connection between orthogonal polynomials and integrable systems.

Contribution

It explicitly links recurrence coefficients of hypergeometric-weight orthogonal polynomials to the discrete Painlevé-V equation and provides a change of variables to transform it into the standard form.

Findings

Recurrence coefficients satisfy discrete Painlevé-V after variable change.

Explicit transformation to standard Painlevé-V form is provided.

Highlights the connection between orthogonal polynomials and integrable systems.

Abstract

Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e equation and determining its type according to Sakai's classification scheme, understanding whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such an example, becomes one of the central ones. Fortunately, Sakai's geometric theory provides an almost algorithmic procedure for answering this question. In this paper we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlev\'e equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for discrete orthogonal polynomials can be expressed in terms of solutions of discrete Painlev\'e equations. In this work we study discrete orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlev\'e-V equation. We also provide an explicit change of variables transforming this equation to the standard form.