Journal of the Mathematical Society of Japan

Integral transformation of Heun's equation and some applications


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It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency of a singularity. For the elliptical representation of Heun's equation, correspondence with respect to monodromy implies isospectral symmetry. We apply the symmetry to finite-gap potentials and express the monodromy of Heun's equation with parameters which have not yet been studied.

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J. Math. Soc. Japan, Volume 69, Number 2 (2017), 849-891.

First available in Project Euclid: 20 April 2017

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Primary: 34M35: Singularities, monodromy, local behavior of solutions, normal forms
Secondary: 33E10: Lamé, Mathieu, and spheroidal wave functions 34M55: Painlevé and other special equations; classification, hierarchies;

Heun's differential equation Euler's integral transformation monodromy Painlevé equation


TAKEMURA, Kouichi. Integral transformation of Heun's equation and some applications. J. Math. Soc. Japan 69 (2017), no. 2, 849--891. doi:10.2969/jmsj/06920849.

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