On reducing the Heun equation to the hypergeometric equation |
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Authors: | Robert S Maier |
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Institution: | Depts. of Mathematics and Physics, University of Arizona, Tucson AZ 85721, USA |
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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. (SIAM J. Math. Anal. 10 (3) (1979) 655). |
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Keywords: | Heun equation Hypergeometric equation Hypergeometric identity Lamé equation Special function Clarkson-Olver transformation |
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