The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

TL;DR

This paper investigates the Stokes phenomenon during the confluence of hypergeometric equation singularities, analyzing how divergence and logarithmic terms relate through Riccati systems and monodromy in complex sectors.

Contribution

It provides a detailed analysis of the Stokes phenomenon in hypergeometric confluence using Riccati equations, linking divergence, logarithmic terms, and monodromy in a unified framework.

Findings

Divergence at irregular singular points explains logarithmic terms at regular points.

Monodromy analysis reveals sector-dependent behavior of solutions.

Stokes multipliers are interpreted as limits of monodromy obstructions.

Abstract

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.