### 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 D_{n}. Each of the 192 expressions is labeled by an element of D_{4}. 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 language | English (US) |
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Pages (from-to) | 811-843 |

Number of pages | 33 |

Journal | Mathematics of Computation |

Volume | 76 |

Issue number | 258 |

DOIs | |

State | Published - Apr 2007 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

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## Cite this

*Mathematics of Computation*,

*76*(258), 811-843. https://doi.org/10.1090/S0025-5718-06-01939-9