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

摘要

The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlev?? equations near infinity. We consider the fifth Painlev?? equation in two cases: when (lphaetagammadelta eq 0) and when (lphaetagamma eq 0), (delta =0) and the third Painlev?? 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 Painlev?? equation and to the third Painlev?? equation, respectively.