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Zbl 1118.34084
Maier, Robert S.
The 192 solutions of the Heun equation. (English)
[J] Math. Comput. 76, No. 258, 811-843 (2007). ISSN 0025-5718; ISSN 1088-6842/e

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 [{\it 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 $ \cal D_n$. Each of the $ 192$ expressions is labeled by an element of $ \cal 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.
[Masaaki Yoshida (Fukuoka)]

MSC 2000:: *34M15 Algebraic aspects of differential equations in the complex domain
33E30 Functions coming from diff., difference and integral equations
33C05 Classical hypergeometric functions
20F55 Coxeter groups
34-04 Machine computation, programs (ordinary differential equations)
33C67 Hypergeometric functions associated with root systems
34M55 Painlevé and other special equations

Citations: ERAM 015.0533cj; ERAM 015.0528cj

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