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On the Convergence of Formal Exotic Series Solutions of an ODE

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Abstract

We propose a sufficient condition for the convergence of a complex power type formal series of the form \(\varphi =\sum _{k=1}^{\infty }\alpha _k(x^{\mathrm{i}\gamma })\,x^k\), where \(\alpha _k\) are functions meromorphic at the origin and \(\gamma \in {{\mathbb {R}}}\setminus \{0\}\), that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of such a type formal solutions of the third Painlevé equation is presented and the proposed sufficient condition is applied to check their convergence; moreover, the accumulation of movable poles of these solutions near the critical point is discussed.

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Notes

  1. There will be another requirement for choosing N, which is not used in the proof of Lemma 1 and which we explain later, in Lemma 2.

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Acknowledgements

We would like to thank Stéphane Malek whose questions and remarks inspired us to consider in more detail the problem of the accumulation of movable poles for local solutions represented by exotic series in the example of the third Painlevé equation.

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Correspondence to R. R. Gontsov.

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Communicated by James K. Langley.

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Gontsov, R.R., Goryuchkina, I.V. On the Convergence of Formal Exotic Series Solutions of an ODE. Comput. Methods Funct. Theory 20, 279–295 (2020). https://doi.org/10.1007/s40315-020-00306-z

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