Geodesic
Power-elliptic expansions of solutions to an ordinary differential equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 12, pp. 2206-2218 Cet article a éte moissonné depuis la source Math-Net.Ru

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A rather general ordinary differential equation is considered that can be represented as a polynomial in variables and derivatives. For this equation, the concept of power-elliptic expansions of its solutions is introduced and a method for computing them is described. It is shown that such expansions of solutions exist for the first and second Painlevé equations. 
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A. D. Bruno. Power-elliptic expansions of solutions to an ordinary differential equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 12, pp. 2206-2218. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_12_a7/

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[2] Bruno A. D., “Space Power geometry for an ODE and Painlevé equations”, Painleve Equations and Related Topics, Internat. Conf. (St. Petersburg, June 2011), 36–41 http://www.pdmi.ras.ru/EIMI/2011/PC/proceedings.pdf

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[5] Bruno A. D., “Space Power geometry try for one ODE and $P_1$–$P_4$, $P_6$”, Painleve Equations and Related Topics, eds. A. D. Bruno, A. B. Batkhin, Walter de Gruyter, Berlin–Boston, 2012, 41–51 | MR

[6] Bruno A. D., Parusnikova A. V., “Elliptic and periodic asymptotic forms of solution to $P_5$”, Painleve Equations and Related Topics, eds. A. D. Bruno, A. B. Batkhin, Walter de Gruyter, Berlin–Boston, 2012, 53–65

[7] Bruno A. D., “Regular asymptotic expansions of solutions to one ODE and $P_1$–$P_5$”, Painleve Equations and Related Topics, eds. A. D. Bruno, A. B. Batkhin, Walter de Gruyter, Berlin–Boston, 2012, 67–82