The 192 solutions of the Heun equation

Research output: Contribution to journalArticle

52 Citations (Scopus)

Abstract

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group Dn. Each of the 192 expressions is labeled by an element of D4. Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group 54. The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

Original languageEnglish (US)
Pages (from-to)811-843
Number of pages33
JournalMathematics of Computation
Volume76
Issue number258
DOIs
StatePublished - Apr 2007

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Heun Equation
Singular Point
Differential equations
Hypergeometric Equation
Coxeter Group
Local Solution
Symmetric group
Automorphism Group
Gauss
Isomorphism
Permutation
Isomorphic
Differential equation

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics
  • Computational Mathematics

Cite this

The 192 solutions of the Heun equation. / Maier, Robert S.

In: Mathematics of Computation, Vol. 76, No. 258, 04.2007, p. 811-843.

Research output: Contribution to journalArticle

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