Orthogonal polynomials: from Heun equations to Painlev\'{e} equations

TL;DR

This paper explores the connection between orthogonal polynomials with various weights, their differential equations, and Painlevé equations, revealing new insights into their asymptotic behaviors and integrable structures.

Contribution

It introduces a method to transform Heun equations associated with orthogonal polynomials into Painlevé equations using isomonodromic deformations, linking recurrence coefficients to integrable systems.

Findings

Heun equations are transformed into Painlevé equations for different weights.

The Painlevé equations match those satisfied by recurrence coefficients.

Asymptotic analysis of Hankel determinants recovers Dyson's constant.

Abstract

In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Derezi\'{n}ski, A. Ishkhanyan, A. Latosi\'{n}ski, SIGMA 17 (2021), 056], we transform these Heun equations into Painlev\'{e} equations. It is interesting that the Painlev\'{e} equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.}