On the Exact Gevrey Order of Formal Puiseux Series Solutions to the Third Painleve Equation

In this paper, we study the third Painleve equation with parameters gamma = 0, alpha delta not equal 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters delta = -beta(2)/2 not equal 0 such that these series are of exact Gevrey order one, and hence diverge. We prove the 1-summability of them and provide analytic functions which are approximated of Gevrey order one by these series in sectors with the vertices at infinity.