On reducing the Heun equation to the hypergeometric equation

Details On reducing the Heun equation to the hypergeometric equation On reducing the Heun equation to the hypergeometric equation

2002-03-26

arxiv.org/abs/math/0203264v4

J. Differential Equations 213 (2005) 171-203

Mathematics

Classical Analysis and ODEs

36 pages, a few additional misprints corrected

Scientific paper

10.1016/j.jde.2004.07.020

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 K. Kuiken, "Heun's equation and the hypergeometric equation", SIAM Journal on Mathematical Analysis 10:3 (1979), 655-657.]

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