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

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 nonlinear differential equations with respect to a well-chosen parameter, according to principles established by D. Chudnovsky and G. Chudnovsky. Examples are given. For instance, the recurrence coefficients in an + 1Pn + 1 (x) = xpn(x) − anpn − 1 (x) of the orthogonal polynomials related to the weight exp (− x44 − tx2) on R satisfy 4an3än = (3an4 + 2tan2 − n)(an4 + 2tan2 + n), and an2 satisfies a Painlevé PIV equation.