On Gevrey orders of formal power series solutions to the third and fifth Painlevé equations near infinity

Abstract

The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painleve equations near infinity. We consider the fifth Painleve equation in two cases: when \(\alpha\beta\gamma\delta \neq 0\) and when \(\alpha\beta\gamma \neq 0\), \(\delta =0\) and the third Painleve equation when all the parameters of the equation are not equal to zero. In the paper we prove Gevrey summability of the formal solutions to the fifth Painleve equation and to the third Painleve equation, respectively.