Transforming the Heun Equation to the Hypergeometric Equation: I. Polynomial Transformations

The reductions of the Heun equation to the hypergeometric equation by rational changes of its independent variable are classified. Heun-to-hypergeometric transformations are analogous to the classical hypergeometric identities (i.e., hypergeometric-to-hypergeometric transformations) of Goursat. However, a transformation is possible only if the singular point location parameter and normalized accessory parameter of the Heun equation are each restricted to take values in a discrete set. The possible changes of variable are all polynomial. They 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.