Global Meromorphy of Solutions of the Painlevé Equations and Their Hierarchies

Abstract

We show that all local holomorphic solutions of all equations constituting the hierarchies of the first and second Painleve equations can be analytically continued to meromorphic functions on the whole complex plane. We also present a new conceptual proof of the fact that all local holomorphic solutions of the first, second, and fourth Painleve equations are globally meromorphic.