Differential equations for the recurrence coefficients of semi-classical orthogonal polynomials and their relation to the Painlev\'e equations via the geometric approach

TL;DR

This paper develops a geometric framework linking differential equations for recurrence coefficients of semi-classical orthogonal polynomials to Painlevé equations, demonstrating the approach with specific examples and exploring related topics.

Contribution

It introduces a general geometric scheme to connect recurrence coefficients of semi-classical orthogonal polynomials with Painlevé equations, extending previous results and analyzing both differential and discrete systems.

Findings

Recurrence coefficients for semi-classical Laguerre polynomials relate to Painlevé IV.

Discrete orthogonal polynomials connect to Painlevé VI.

The geometric approach reveals additional considerations in initial condition construction.

Abstract

In this paper we present a general scheme for how to relate differential equations for the recurrence coefficients of semi-classical orthogonal polynomials to the Painlev\'e equations using the geometric framework of the Okamoto Space of Initial Conditions. We demonstrate this procedure in two examples. For semi-classical Laguerre polynomials appearing in \cite{HC17}, we show how the recurrence coefficients are connected to the fourth Painlev\'e equation. For discrete orthogonal polynomials associated with the hypergeometric weight appearing in \cite{FVA18} we discuss the relation of the recurrence coefficients to the sixth Painlev\'e equation, extending the results of \cite{DFS19}, where a similar approach was used for a discrete system for the same recurrence coefficients. Though the discrete and differential systems here share the same geometry, the construction of the space of initial conditions from the differential system is different and reveals extra considerations that must be made. We also discuss a number of related topics in the context of the geometric approach, such as Hamiltonian forms of the differential equations for the recurrence coefficients, Riccati solutions for special parameter values, and associated discrete Painlev\'e equations.