The 192 Solutions of the Heun Equation

TL;DR

This paper provides a comprehensive list of 192 local solutions to the Heun equation, analyzing their symmetries and group structures, and drawing parallels to classical hypergeometric solutions.

Contribution

It introduces a machine-generated enumeration of all local solutions of the Heun equation and explores their automorphism group structure, linking it to Coxeter groups and symmetric groups.

Findings

192 solutions classified and labeled by D_4

24 solutions equivalent to the local Heun function Hl

Transformation group of Hl is isomorphic to S_4

Abstract

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D_n. Each of the 192 expressions is labeled by an element of D_4. Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S_4. The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.