Stokes matrices for a class of reducible equations

TL;DR

This paper explicitly computes Stokes matrices for a class of reducible differential equations at irregular singularities, extending previous work with new methods and broader parameter conditions.

Contribution

It provides explicit calculations of Stokes matrices for more general equations using Borel-Laplace summation and iterated integrals, under less restrictive assumptions.

Findings

Computed Stokes matrices for three families of equations.

Extended validity of results to specific parameter cases.

Provided explicit representation of 1-sums of divergent series.

Abstract

This paper is a continuation of our previous work \cite{St} where we have studied the Stokes phenomenon for a particular family of equation \eqref{initial} with \eqref{form-0}-\eqref{npe} from a perturbative point of view. Here we focus on the explicit computation of the Stokes matrices at the non-resonant irregular singularity for a more general situation. In particular, utilizing Borel-Laplace summation, the iterated integrals approach and some properties of the hypergeometric series we compute by hand the Stokes matrices of three families of equation \eqref{initial}-\eqref{form-0}-\eqref{npe} under assumptions that $\beta_j$'s are distinct and $|\beta_3-\beta_1| < |\beta_3-\beta_2|$. Moreover, these results remain valid for these distinct $\beta_j$'s for which $|\beta_3-\beta_1|=|\beta_3-\beta_2|$ but $\beta_3-\beta_1 \neq \pm (\beta_3-\beta_2)$ on condition that $\mathcal{Re} (\alpha_2-\alpha_1) > -1$. In addition, iterated integrals approach allows us to give, under some restrictions, an explicit representation of the 1-sum of the product of two certain divergent 1-summable power series, that have different singular directions.