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Fuchsian equation, Hermite-Krichever Ansatz and Painlevé equation

Kouichi Takemura
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Abstract

Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed.

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Kouichi Takemura. 2005-01-25. Fuchsian equation, Hermite-Krichever Ansatz and Painlevé equation. https://arxiv.org/abs/math/0501428

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