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Integral transformation of Heun’s equation and some applications. (English) Zbl 1371.34132

Author’s abstract: 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.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
33E10 Lamé, Mathieu, and spheroidal wave functions
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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