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In my research I have come across a divergent asymptotic series $\sum_{n =0}^\infty a_n f_n(x)$ that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond to any standard special functions.

Here is my question: Given such an asymptotic series, what are the standard methods for proving that a function exists that has that asymptotic expansion around infinity and solves the relevant ODE?

This seems like it should be a classical topic (and is presumably related to Borel resummation etc), so I was surprised I could not really find anything online. Perhaps I'm missing the relevant key words?

I have avoided writing the specific ODE and divergent series since I would mostly like to learn about the general techniques!

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As I recall, this book deals with it:

Costin, Ovidiu, Asymptotics and Borel summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 141. Boca Raton, FL: Chapman & Hall/CRC (ISBN 978-1-4200-7031-6/hbk). xiii, 250 p. (2009). ZBL1169.34001.

Assorted papers are found here:

Ledoux, M. (ed.), Proceedings of the conference on “Resurgence, alien calculus, resummability, transseries”, in honour of J. Ecalle, November 18–22, 2002 at CIRM (Centre International de Rencontres Mathématiques), Marseille, France, Ann. Fac. Sci. Toulouse, Math. (6) 13, No. 3, 289-475 (2004); No. 4, 477-708 (2004). ZBL1157.00321.

Ecalle spent many years working on this. It started with his efforts on Hilbert's 16th problem. The desired solutions were found as asymptotic series, which (it turned out) may not converge; but in some sense they still do represent the solutions we want. Ecalle worked to answer the question: In what sense?

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