Properties of the series solution for Painleve I

TL;DR

This paper investigates the asymptotic behavior of Laurent series coefficients for solutions of Painleve I, providing recursive formulas for the tau-function and analyzing specific solutions.

Contribution

It introduces explicit recursive formulas for the tau-function's Taylor expansion and explores the asymptotics of solutions, extending classical elliptic function results.

Findings

Recursive formulas for tau-function expansion

Asymptotic behavior of Laurent series coefficients

Analysis of symmetric solution at the origin

Abstract

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.