Stokes matrices for a class of reducible equations

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.