Fuchsian equation, Hermite-Krichever Ansatz and Painlevé equation

Details Fuchsian equation, Hermite-Krichever Ansatz and Painlevé equation Fuchsian equation, Hermite-Krichever Ansatz and Painlevé equation

2005-01-25

arxiv.org/abs/math/0501428v1

Mathematics

Classical Analysis and ODEs

28 pages

Scientific paper

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 Painlev\'e equation by monodromy preserving deformation and obtain solutions of the sixth Painlev\'e equation which include Hitchin's solution. The relationship with finite-gap potential is also discussed.

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