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

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 $z^{2\mathrm{/}3}$ – 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.