## The 192 solutions of the Heun equation

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- by Robert S. Maier;
- Math. Comp.
**76**(2007), 811-843 - DOI: https://doi.org/10.1090/S0025-5718-06-01939-9
- Published electronically: November 28, 2006
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## Abstract:

A machine-generated list of $192$ local solutions of the Heun equation is given. They are analogous to Kummerâ€™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.## References

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, DC, 1964. For sale by the Superintendent of Documents. MR**167642** - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - M. Baake,
*Structure and representations of the hyperoctahedral group*, J. Math. Phys.**25**(1984), no.Â 11, 3171â€“3182. MR**761836**, DOI 10.1063/1.526087 - Francesco Brenti,
*$q$-Eulerian polynomials arising from Coxeter groups*, European J. Combin.**15**(1994), no.Â 5, 417â€“441. MR**1292954**, DOI 10.1006/eujc.1994.1046 - B. Dwork,
*On Kummerâ€™s twenty-four solutions of the hypergeometric differential equation*, Trans. Amer. Math. Soc.**285**(1984), no.Â 2, 497â€“521. MR**752488**, DOI 10.1090/S0002-9947-1984-0752488-X - A. ErdĂ©lyi, editor.
*Higher Transcendental Functions*. McGrawâ€“Hill, New York, 1953â€“55. Also known as*The Bateman Manuscript Project*. - K. Franz.
*Untersuchungen ĂĽber die lineare homogene Differentialgleichung 2. Ordnung der Fuchsâ€™schen Klasse mit drei im Endlichen gelegenen singulĂ¤ren Stellen*. Inaugural dissertation, Friedrichs-UniversitĂ¤t Halle-Wittenberg, 1898. - F. Gesztesy and R. Weikard,
*Treibich-Verdier potentials and the stationary (m)KdV hierarchy*, Math. Z.**219**(1995), no.Â 3, 451â€“476. MR**1339715**, DOI 10.1007/BF02572375 - Jeremy J. Gray,
*Linear differential equations and group theory from Riemann to PoincarĂ©*, 2nd ed., BirkhĂ¤user Boston, Inc., Boston, MA, 2000. MR**1751835** - L. C. Grove and C. T. Benson,
*Finite reflection groups*, 2nd ed., Graduate Texts in Mathematics, vol. 99, Springer-Verlag, New York, 1985. MR**777684**, DOI 10.1007/978-1-4757-1869-0 - K. Heun. Zur Theorie der Riemannâ€™schen Functionen zweiter Ordnung mit vier Verzweigungspunkten.
*Math. Ann.*, 33:161â€“179, 1889. - E. L. Ince,
*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**10757** - E. E. Kummer. Ăśber die hypergeometrische Reihe $1 + \frac {\alpha .\beta } {1.\gamma } x + \frac {\alpha (\alpha +1)\beta (\beta +1)} {1.2.\gamma (\gamma +1)} x^2 + \frac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)} {1.2.3.\gamma (\gamma +1)(\gamma +2)} x^3 + \ldots$
*J. Reine Angew. Math.*, 15:39â€“83, 127â€“172, 1836. - S.-T. Ma.
*Relations Between the Solutions of a Linear Differential Equation of Second Order with Four Regular Singular Points*. Ph.D. dissertation, University of California, Berkeley, Dept. of Mathematics, 1934. - Robert S. Maier,
*On reducing the Heun equation to the hypergeometric equation*, J. Differential Equations**213**(2005), no.Â 1, 171â€“203. MR**2139342**, DOI 10.1016/j.jde.2004.07.020 - S. V. Oblezin,
*Discrete symmetries of systems of isomonodromic deformations of second-order differential equations of Fuchsian type*, Funktsional. Anal. i Prilozhen.**38**(2004), no.Â 2, 38â€“54, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl.**38**(2004), no.Â 2, 111â€“124. MR**2086626**, DOI 10.1023/B:FAIA.0000034041.67089.07 - E. G. C. Poole.
*Linear Differential Equations*. Oxford University Press, Oxford, 1936. - Reese T. Prosser,
*On the Kummer solutions of the hypergeometric equation*, Amer. Math. Monthly**101**(1994), no.Â 6, 535â€“543. MR**1274975**, DOI 10.2307/2975319 - A. Ronveaux (ed.),
*Heunâ€™s differential equations*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by F. M. Arscott, S. Yu. Slavyanov, D. Schmidt, G. Wolf, P. Maroni and A. Duval. MR**1392976** - A. Ronveaux,
*Factorization of the Heunâ€™s differential operator*, Appl. Math. Comput.**141**(2003), no.Â 1, 177â€“184. Advanced special functions and related topics in differential equations (Melfi, 2001). MR**1986079**, DOI 10.1016/S0096-3003(02)00331-4 - Reinhard SchĂ¤fke and Dieter Schmidt,
*The connection problem for general linear ordinary differential equations at two regular singular points with applications in the theory of special functions*, SIAM J. Math. Anal.**11**(1980), no.Â 5, 848â€“862. MR**586913**, DOI 10.1137/0511076 - F. Schmitz and B. Fleck. On the propagation of linear 3-D hydrodynamic waves in plane non-isothermal atmospheres.
*Astron. Astrophys. Suppl. Ser.*, 106(1):129â€“139, 1994. - Alexander O. Smirnov,
*Elliptic solitons and Heunâ€™s equation*, The Kowalevski property (Leeds, 2000) CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp.Â 287â€“305. MR**1916788**, DOI 10.1090/crmp/032/16 - Alexander O. Smirnov,
*Finite-gap solutions of the Fuchsian equations*, Lett. Math. Phys.**76**(2006), no.Â 2-3, 297â€“316. MR**2238723**, DOI 10.1007/s11005-006-0070-x - Chester Snow,
*Hypergeometric and Legendre functions with applications to integral equations of potential theory*, National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, DC, 1952. MR**48145** - V. S. Varadarajan,
*Linear meromorphic differential equations: a modern point of view*, Bull. Amer. Math. Soc. (N.S.)**33**(1996), no.Â 1, 1â€“42. MR**1339809**, DOI 10.1090/S0273-0979-96-00624-6 - Virginia W. Wakerling,
*The relations between solutions of the differential equation of the second order with four regular singular points*, Duke Math. J.**16**(1949), 591â€“599. MR**35347** - E. T. Whittaker and G. N. Watson,
*A course of modern analysis*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR**1424469**, DOI 10.1017/CBO9780511608759 - Masaaki Yoshida,
*A presentation of the fundamental group of the configuration space of $5$ points on the projective line*, Kyushu J. Math.**48**(1994), no.Â 2, 283â€“289. MR**1294531**, DOI 10.2206/kyushujm.48.283

## Bibliographic Information

**Robert S. Maier**- Affiliation: Departments of Mathematics and Physics, University of Arizona, Tucson, Arizona 85721
- MR Author ID: 118320
- ORCID: 0000-0002-1259-1341
- Email: rsm@math.arizona.edu
- 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.
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**76**(2007), 811-843 - MSC (2000): Primary 33E30; Secondary 33-04, 34M15, 33C05, 20F55
- DOI: https://doi.org/10.1090/S0025-5718-06-01939-9
- MathSciNet review: 2291838