On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem

Abstract

We propose a sufficient condition of the convergence of a complex power type formal series of the form $\phi ={\sum }_{k=1}^{\infty }{\alpha }_k(x^{\mathrm{i}\gamma })x^k$, 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 a such type formal solution of the third Painlevé equation is presented and the proposed sufficient condition is applied to check its convergence.

13 pages