Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials.

Abstract

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w^{\prime }/w$ 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 $a_{n+1}{p}_{n+1}(x)=x{p}_n(x)-a_n{p}_{n-1}(x)$ of the orthogonal polynomials related to the weight $\exp (-x^4/4-t{x}^2)$ on {\blackb R\/} satisfy $4a_n^3{\ddot{a}}_n=(3a_n^4+2t{a}_n^2-n)(a_n^4+2t{a}_n^2+n)$, and $a_n^2$ satisfies a Painlevé ${\mathrm{P}}_{\mathrm{IV}}$ equation.