On Reducing the Heun Equation to the Hypergeometric Equation

The reductions of the Heun equation to the hypergeometric equation by rational transformations of its independent variable are enumerated and classified. Heun-tohypergeometric 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 possible reductions all involve polynomial transformations. These 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.]