Stokes phenomenon arising in the confluence of the Gauss hypergeometric equation

TL;DR

This paper explores the Stokes phenomenon in the confluence of Gauss hypergeometric equations, deriving explicit Stokes matrices from monodromy data as singularities merge.

Contribution

It provides a detailed analysis of the confluence process, linking solutions' asymptotic behaviors to monodromy and connection matrices, with explicit calculations of Stokes matrices.

Findings

Explicit formulas for Stokes matrices in confluent hypergeometric systems

Connection between monodromy data and asymptotic solution behavior

Analysis of solution behavior transition from power-like to exponential

Abstract

In this paper we study the Gauss and Kummer hypergeometric equations in depth. In particular, we focus on the confluence of two regular singularities of the Gauss hypergeometric equation to produce the Kummer hypergeometric equation with an irregular singularity at infinity. We show how to pass from solutions with power-like behaviour which are analytic in disks, to solutions with exponential behaviour which are analytic in sectors and have divergent asymptotics. We explicitly calculate the Stokes matrices of the confluent system in terms of the monodromy data, specifically the connection matrices, of the original system around the merging singularities.