Rational solutions to the Painleve II equation from particular polynomials
Résumé
The Painleve equations were derived by Painleve and Gambier in 1895-1910. Given a rational function R in w, w' and analytic in z, they searched what were the second order ordinary differential equations of the form w '' = R(z, w, w') with the properties that the singularities other than poles of any solution or this equation depend on the equation only and not of the constants of integration. They proved that there are 50 equations of this type, and the Painleve II is one of these. Here, we construct solutions to the Painleve II equation (PII) from particular polynomials. We obtain rational solutions written as a derivative with respect to the variable x of a logarithm of a quotient of a determinant of order n + 1 by a determinant of order n. We obtain an infinite hierarchy of rational solutions to the PII equation. We give explicitly the expressions of these solutions solution for the first orders.