On reducing the Heun equation to the hypergeometric equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:213

Article

Maier, RS

关键词: Heun equation;    hypergeometric equation;    hypergeometric identity;    Lame equation;    special function;    Clarkson-Olver transformation;   

DOI  :  10.1016/j.jde.2004.07.020

来源: Elsevier

【 摘 要 】

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. (c) 2004 Elsevier Inc. All rights reserved.

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