Generalized Okubo systems and the middle convolution

Abstract

The present thesis consists of the two part:/ I Generalized Okubo systems and the middle convolution / II Confluence of singular points and the Okubo systems In the first part, we give a generalization, called a generalized Okubo system, of a system of linear differential equations of the Okubo normal form and define a mapping π from a set of generalized Okubo systems to a set of linear differential systems. We consider the operation, the middle convolution introduced by Katz, using π, and show that any system of linear differential equations, not necessarily of the Fuchsian type, with a regular singularity at infinity, can be transformed into generalized Okubo system. For any non-Fuchsian system, we can construct a Fuchsian system with a parameter ε which tends to the given equation as ε→ 0. In the second part, we consider a confluence of the convolution in the sense of Katz-Dettweiler-Reiter and we show that convolutions of each equation is compatiple with the confluence.