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The 192 solutions of the Heun equation. (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 (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain 33E30 Other functions coming from differential, difference and integral equations 33C05 Classical hypergeometric functions, $${}_2F_1$$ 20F55 Reflection and Coxeter groups (group-theoretic aspects) 34-04 Software, source code, etc. for problems pertaining to ordinary differential equations 33C67 Hypergeometric functions associated with root systems 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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