Integral transformation of Heun's equation and some applications

• Takemura Kouichi

  Department of Mathematics, Faculty of Science and Engineering, Chuo University

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.

Journal

• Journal of the Mathematical Society of Japan

  Journal of the Mathematical Society of Japan 69 (2), 849-891, 2017

  The Mathematical Society of Japan

Keywords

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

Details

• CRID: 1390001205115193856
• NII Article ID: 130006887125
• NII Book ID: AA0070177X
• DOI: 10.2969/jmsj/06920849
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL BIB ID: 028132597
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I028132597
• Text Lang: en
• Data Source:
  • JaLC
  • NDL
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed