# On Divergence of Puiseux Series Asymptotic Expansions of Solutions to the Third Painlev\'{e} Equation

## TL;DR

This paper investigates the divergence of Puiseux series solutions to the third Painlevé equation, identifying parameter conditions where these series are Gevrey order one and approximating solutions in sectors at infinity.

## Contribution

It establishes the divergence of specific Puiseux series solutions and constructs analytic functions approximated by these series in sectors at infinity.

## Key findings

- Puiseux series are of exact Gevrey order one under certain parameters.
- The series diverge, but approximate solutions are constructed in sectors.
- Provides insight into the asymptotic behavior of solutions at infinity.

## Abstract

In this paper we present a family of values of the parameters of the third Painlev\'{e} equation such that Puiseux series formally satisfying this equation -- considered as series of $z^{2/3}$ -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.