### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

**On reducing the Heun equation to the hypergeometric equation.** / Maier, Robert S.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 213, no. 1, pp. 171-203. https://doi.org/10.1016/j.jde.2004.07.020

}

TY - JOUR

T1 - On reducing the Heun equation to the hypergeometric equation

AU - Maier, Robert S

PY - 2005/6/1

Y1 - 2005/6/1

N2 - 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.

AB - 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.

KW - Clarkson-Olver transformation

KW - Heun equation

KW - Hypergeometric equation

KW - Hypergeometric identity

KW - Lamé equation

KW - Special function

UR - http://www.scopus.com/inward/record.url?scp=18144366618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18144366618&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2004.07.020

DO - 10.1016/j.jde.2004.07.020

M3 - Article

AN - SCOPUS:18144366618

VL - 213

SP - 171

EP - 203

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -