Hirota bilinear equations for Painlev\'e transcendents

Abstract

We present some observations on the tau-function for the fourth Painlev\'e equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlev\'e equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.