Title: 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.