Hirota bilinear equations for Painlevé transcendents

Abstract

We present some observations on the tau-function for the fourth Painlevé 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 Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

This is version n.2, some typos have been fixed (e.g. formula 31), a formula for the coefficients \hat{A}_j in section 4 has been added, the references have been updated