Painlevé Property and Generating Functions for Asymptotics

This paper proposes a new approach to the asymptotic analysis of Painlevé equations. The approach is based on representing solutions of the Painlevé equations using formal series in two variables, ${\sum }_{k=0}^{\mathrm{\infty }}{y}^k{A}_k(x)$, with rational functions $A_k(x)$. The approach is applied to the asymptotic analysis of the third degenerate Painlevé equation.