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Zbl 1085.34035
Maier, Robert S.
On reducing the Heun equation to the hypergeometric equation. (English)
[J] J. Differ. Equations 213, No. 1, 171-203 (2005). ISSN 0022-0396

Summary: 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; see {\it K. Kuiken} [SIAM J. Math. Anal. 10, 655--657 (1979; Zbl 0415.34038)].

MSC 2000:: *34C20 Transformation of ODE and systems

Keywords: Heun equation; Hypergeometric equation; Hypergeometric identity; Lamé equation; Special function; Clarkson-Olver transformation

Citations: Zbl 0415.34038

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