On reducing the Heun equation to the hypergeometric equation

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57 Scopus citations

Abstract

The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations.

Original languageEnglish (US)
Pages (from-to)171-203
Number of pages33
JournalJournal of Differential Equations
Volume213
Issue number1
DOIs
StatePublished - Jun 1 2005

Keywords

  • Clarkson-Olver transformation
  • Heun equation
  • Hypergeometric equation
  • Hypergeometric identity
  • Lamé equation
  • Special function

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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