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The 192 solutions of the Heun equation

Author: Robert S. Maier
Journal: Math. Comp. 76 (2007), 811-843
MSC (2000): Primary 33E30; Secondary 33-04, 34M15, 33C05, 20F55
Published electronically: November 28, 2006
MathSciNet review: 2291838
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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  $ \mathop{{}\it Hl}\nolimits $, and it is shown that the resulting order-$ 24$ group of transformations of  $ \mathop{{}\it Hl}\nolimits $ 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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Robert S. Maier
Affiliation: Departments of Mathematics and Physics, University of Arizona, Tucson, Arizona 85721

Received by editor(s): August 23, 2004
Received by editor(s) in revised form: February 7, 2006
Published electronically: November 28, 2006
Additional Notes: The author was supported in part by NSF Grant No. PHY-0099484.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.