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


SIGMA 5 (2009), 040, 22 pages      arXiv:0810.3112      http://dx.doi.org/10.3842/SIGMA.2009.040
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Middle Convolution and Heun's Equation

Kouichi Takemura
Department of Mathematical Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan

Received November 26, 2008, in final form March 25, 2009; Published online April 03, 2009

Abstract
Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of initial conditions of the sixth Painlevé equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

Key words: Heun's equation; the space of initial conditions; the sixth Painlevé equation; middle convolution.

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