*(English)*Zbl 1118.34084

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 [*E. E. Kummer*, J. Reine Angew. Math. 15, 39–83, 127–172 (1836; ERAM 015.0528cj and ERAM 015.0533cj)], 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 ${\mathcal{D}}_{n}$. Each of the 192 expressions is labeled by an element of ${\mathcal{D}}_{4}$.

For the group of order 192, the structure and the action are neatly explained. Each solution is expressed as the product of complex powers of $x$, $1-x$, $a-x$, and the function $Hl$ with different parameters and variable.

There are 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.

##### MSC:

34M15 | Algebraic aspects of ODE in the complex domain |

33E30 | Functions coming from differential, difference and integral equations |

33C05 | Classical hypergeometric functions, ${}_{2}{F}_{1}$ |

20F55 | Reflection groups; Coxeter groups |

34-04 | Machine computation, programs (ordinary differential equations) |

33C67 | Hypergeometric functions associated with root systems |

34M55 | Painlevé and other special equations; classification, hierarchies |