### 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 language | English (US) |
---|---|

Pages (from-to) | 171-203 |

Number of pages | 33 |

Journal | Journal of Differential Equations |

Volume | 213 |

Issue number | 1 |

DOIs | |

State | Published - Jun 1 2005 |

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics