This article is cited in 7 papers

A Riemann–Hilbert Approach to the Heun Equation

Boris Dubrovin^a, Andrei Kapaev<sup>b

a</sup> SISSA, Via Bonomea 265, 34136, Trieste, Italy
^b Deceased

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.

Keywords: Heun polynomials; Riemann–Hilbert problem; Painlevé equations.

MSC: 34M03; 34M05; 34M35; 34M55; 57M50

Received: February 7, 2018; in final form August 15, 2018; Published online September 7, 2018

Language: English

DOI: 10.3842/SIGMA.2018.093

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