×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. · Zbl 0171.38503
[2] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. · Zbl 0920.33001
[3] M. Baake, Structure and representations of the hyperoctahedral group, J. Math. Phys. 25 (1984), no. 11, 3171 – 3182. · Zbl 0551.20008
[4] Francesco Brenti, \?-Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), no. 5, 417 – 441. · Zbl 0809.05012
[5] B. Dwork, On Kummer’s twenty-four solutions of the hypergeometric differential equation, Trans. Amer. Math. Soc. 285 (1984), no. 2, 497 – 521. · Zbl 0552.12012
[6] A. Erdélyi, editor. Higher Transcendental Functions. McGraw-Hill, New York, 1953-55. Also known as The Bateman Manuscript Project.
[7] K. Franz. Untersuchungen über die lineare homogene Differentialgleichung 2. Ordnung der Fuchs’schen Klasse mit drei im Endlichen gelegenen singulären Stellen. Inaugural dissertation, Friedrichs-Universität Halle-Wittenberg, 1898. · JFM 30.0311.02
[8] F. Gesztesy and R. Weikard, Treibich-Verdier potentials and the stationary (m)KdV hierarchy, Math. Z. 219 (1995), no. 3, 451 – 476. · Zbl 0830.35119
[9] Jeremy J. Gray, Linear differential equations and group theory from Riemann to Poincaré, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2000. · Zbl 0949.01001
[10] L. C. Grove and C. T. Benson, Finite reflection groups, 2nd ed., Graduate Texts in Mathematics, vol. 99, Springer-Verlag, New York, 1985. · Zbl 0579.20045
[11] K. Heun. Zur Theorie der Riemann’schen Functionen zweiter Ordnung mit vier Verzweigungspunkten. Math. Ann., 33:161-179, 1889.
[12] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. · Zbl 0063.02971
[13] E. E. Kummer. Über die hypergeometrische Reihe \( 1 + \frac {\alpha.\beta} {1.\gamma} x + \frac {\alpha(\alpha+1)\beta(\beta+1)}... ...lpha+2)\beta(\beta+1)(\beta+2)\) {1.2.3.\gamma(\gamma+1)(\gamma+2)} x^3 + \ldots\( J. Reine Angew. Math., 15:39-83, 127-172, 1836.\) · ERAM 015.0528cj
[14] 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.
[15] Robert S. Maier, On reducing the Heun equation to the hypergeometric equation, J. Differential Equations 213 (2005), no. 1, 171 – 203. · Zbl 1085.34035
[16] 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. 2, 38 – 54, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 38 (2004), no. 2, 111 – 124. · Zbl 1084.32010
[17] E. G. C. Poole. Linear Differential Equations. Oxford University Press, Oxford, 1936. · Zbl 0014.05801
[18] Reese T. Prosser, On the Kummer solutions of the hypergeometric equation, Amer. Math. Monthly 101 (1994), no. 6, 535 – 543. · Zbl 0813.33001
[19] A. Ronveaux , Heun’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. · Zbl 0847.34006
[20] A. Ronveaux, Factorization of the Heun’s differential operator, Appl. Math. Comput. 141 (2003), no. 1, 177 – 184. Advanced special functions and related topics in differential equations (Melfi, 2001). · Zbl 1035.34100
[21] Reinhard Schäfke 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. 5, 848 – 862. , https://doi.org/10.1137/0511076 Reinhard Schäfke, The connection problem for two neighboring regular singular points of general linear complex ordinary differential equations, SIAM J. Math. Anal. 11 (1980), no. 5, 863 – 875. · Zbl 0443.34026
[22] 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):129-139, 1994.
[23] Alexander O. Smirnov, Elliptic solitons and Heun’s equation, The Kowalevski property (Leeds, 2000) CRM Proc. Lecture Notes, vol. 32, Amer. Math. Soc., Providence, RI, 2002, pp. 287 – 305. · Zbl 1072.34102
[24] Alexander O. Smirnov, Finite-gap solutions of the Fuchsian equations, Lett. Math. Phys. 76 (2006), no. 2-3, 297 – 316. · Zbl 1136.34065
[25] 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, D. C., 1952. · Zbl 0048.04702
[26] V. S. Varadarajan, Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 1 – 42. · Zbl 0862.34004
[27] 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), 591 – 599. · Zbl 0035.17601
[28] 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. · JFM 53.0180.04
[29] 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. 2, 283 – 289. · Zbl 0847.57001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.