The 192 solutions of the Heun equation

Maier, Robert

doi:10.1090/S0025-5718-06-01939-9

The 192 solutions of the Heun equation
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Math. Comp. 76 (2007), 811-843 Request permission

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 $\mathcal {D}_n$. Each of the $192$ expressions is labeled by an element of $\mathcal {D}_4$. 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.

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