Confluence of singularities in hypergeometric systems

TL;DR

This paper explores the confluence of singularities in hypergeometric systems, linking Birkhoff and Okubo forms through limiting procedures and analyzing their solutions and monodromy properties.

Contribution

It establishes a connection between Birkhoff and Okubo systems via a limiting process, providing a new perspective on their solutions and monodromy relations.

Findings

Confluence relates monodromy and Stokes matrices.

Solutions of Okubo systems converge to canonical solutions of Birkhoff systems.

Illustrated with generalized hypergeometric equations.

Abstract

A system in a Birkhoff normal form with an irregular singularity of Poincare rank 1 at the origin and a regular singularity at infinity is through the Borel-Laplace transform dual to a system in an Okubo form. Schafke has showed that the Birkhoff system can also be obtained from the Okubo system by a simple limiting procedure. The Okubo system comes naturally with two kinds of mixed solution bases, both of which converge under the limit procedure to the canonical solutions of the limit Birkhoff system on sectors near the irregular singularity at the origin. One can then define Stokes matrices of the Okubo system as connection matrices between different branches of the mixed solution bases and use them to relate the monodromy matrices of the Okubo system to the usual Stokes matrices of the limit system at the irregular singularity. This is illustrated on the example of confluence in the generalized hypergeometric equation.