On the Convergence of Formal Exotic Series Solutions of an ODE

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 Painleve 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.