Painlev\'e-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.

TL;DR

This paper demonstrates that recurrence coefficients of semi-classical orthogonal polynomials satisfy Painlevé-type differential equations, providing explicit examples and linking these coefficients to well-known integrable systems.

Contribution

It establishes a connection between recurrence coefficients of semi-classical orthogonal polynomials and Painlevé equations, extending the understanding of their differential properties.

Findings

Recurrence coefficients satisfy nonlinear differential equations.

Explicit example with weight exp(-x^4/4 - t x^2).

Recurrence coefficient squared satisfies Painlevé IV.

Abstract

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect to a well-chosen parameter, according to principles established by D. G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in $a_{n+1}p_{n+1}(x)=xp_n(x) -a_np_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp(-x^4/4-tx^2)$ on {\blackb R\/} satisfy $4a_n^3\ddot a_n = (3a_n^4+2ta_n^2-n)(a_n^4+2ta_n^2+n)$, and $a_n^2$ satisfies a Painlev\'e ${\rm P}_{\rm IV}$ equation.