Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 093, 24 pages      arXiv:1809.02311      https://doi.org/10.3842/SIGMA.2018.093
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

A Riemann-Hilbert Approach to the Heun Equation

Boris Dubrovin a and Andrei Kapaev b
a) SISSA, Via Bonomea 265, 34136, Trieste, Italy
b) Deceased

Received February 07, 2018, in final form August 15, 2018; Published online September 07, 2018

Abstract
We describe the close connection between the linear system for the sixth Painlevé 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.

Key words: Heun polynomials; Riemann-Hilbert problem; Painlevé equations.

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