A Riemann-Hilbert Approach to the Heun Equation

TL;DR

This paper explores the connection between the sixth Painlevé equation and the Heun equation through a Riemann-Hilbert framework, enabling explicit polynomial solutions in special cases.

Contribution

It introduces a Riemann-Hilbert approach to analyze the Heun equation, linking it to Painlevé equations and providing a method to construct solutions.

Findings

Established the link between Painlevé VI and Heun equations.

Formulated the Riemann-Hilbert problem for Heun functions.

Constructed explicit polynomial solutions for reducible monodromy cases.

Abstract

We describe the close connection between the linear system for the sixth Painlev\'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.