Introduction to middle convolution for differential equations with irregular singularities

TL;DR

This paper introduces middle convolution for linear differential equations with irregular singularities, proposes a definition of their rigidity index, and classifies terminal patterns of irreducible systems using this method.

Contribution

It extends middle convolution to irregular singularities, providing a new framework for analyzing the structure and classification of such differential systems.

Findings

Defined the index of rigidity for irregular singularities

Classified terminal patterns of irreducible systems using middle convolution

Established conditions under which the classification applies

Abstract

We introduce middle convolution for systems of linear differential equations with irregular singular points, and we presend a tentative definition of the index of rigidity for them. Under some assumption, we show a list of terminal patterns of irreducible systems of linear differential equations by iterated application of middle convolution when the index is positive or zero.