### 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) |
---|---|

Pages (from-to) | 811-843 |

Number of pages | 33 |

Journal | Mathematics of Computation |

Volume | 76 |

Issue number | 258 |

DOIs | |

State | Published - Apr 2007 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

*Mathematics of Computation*,

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

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

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 76, no. 258, pp. 811-843. https://doi.org/10.1090/S0025-5718-06-01939-9

}

TY - JOUR

T1 - The 192 solutions of the Heun equation

AU - Maier, Robert S

PY - 2007/4

Y1 - 2007/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34948831246&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34948831246&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-06-01939-9

DO - 10.1090/S0025-5718-06-01939-9

M3 - Article

VL - 76

SP - 811

EP - 843

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 258

ER -