On the Convergence of Formal Exotic Series Solutions of an ODE

We propose a sufficient condition for the convergence of a complex power type formal series of the form φ=∑k=1∞αk(xiγ)xk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi =\sum _{k=1}^{\infty }\alpha _k(x^{\mathrm{i}\gamma })\,x^k$$\end{document}, where αk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _k$$\end{document} are functions meromorphic at the origin and γ∈R\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in {{\mathbb {R}}}\setminus \{0\}$$\end{document}, 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.