Takemura, Kouichi Integral transformation of Heun’s equation and some applications. (English) Zbl 1371.34132 J. Math. Soc. Japan 69, No. 2, 849-891 (2017). 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. Reviewer: Vladimir P. Kostov (Nice) Cited in 2 Documents 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 Keywords:Heun’s differential equation; Euler’s integral transformation; monodromy; Painlevé equation PDFBibTeX XMLCite \textit{K. Takemura}, J. Math. Soc. Japan 69, No. 2, 849--891 (2017; Zbl 1371.34132) Full Text: DOI arXiv Euclid