Padé Method to Painlevé Equations

Abstract

A class of special solutions of Painlevé/Garnier systems arising as the Bäcklund or Schlesinger transformations of the Riccati solutions is known. In the past several years, the corresponding τ-functions have been explicitly computed and expressed as certain specialization of the Schur functions with rectangle shape partitions. In this note, we will give a simple and direct derivation of these solutions. Our method is based on the Padé approximation and its intrinsic relation to iso-monodromy deformations.