Opuscula Math. 34, no. 3 (2014), 591-599
http://dx.doi.org/10.7494/OpMath.2014.34.3.591

Opuscula Mathematica

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

Anastasia V. Parusnikova

Abstract. The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlevé equations near infinity. We consider the fifth Painlevé equation in two cases: when $$\alpha\beta\gamma\delta \neq 0$$ and when $$\alpha\beta\gamma \neq 0$$, $$\delta =0$$ and the third Painlevé 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 Painlevé equation and to the third Painlevé equation, respectively.

Keywords: Painlevé equations, Newton polygon, asymptotic expansions, Gevrey orders.

Mathematics Subject Classification: 34M25, 34M55.

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• Anastasia V. Parusnikova
• National Research University Higher School of Economics, Bolshoi Trekhsvjatitelskii per. 3, Moscow, 109028, Russia
• Revised: 2014-02-22.
• Accepted: 2014-03-21.