Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 056, 59 pages      arXiv:2007.05698

From Heun Class Equations to Painlevé Equations

Jan Dereziński a, Artur Ishkhanyan bc and Adam Latosiński a
a) Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093, Warszawa, Poland
b) Russian-Armenian University, 0051 Yerevan, Armenia
c) Institute for Physical Research of NAS of Armenia, 0203 Ashtarak, Armenia

Received August 25, 2020, in final form May 25, 2021; Published online June 07, 2021

In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.

Key words: linear ordinary differential equation; Heun class equations; isomonodromy deformations; Painlevé equations.

pdf (702.9 KB)   tex (51.6 KB)  


  1. Chiang Y.-M., Law C.-K., Yu G.-F., Invariant subspaces of biconfluent Heun operators and special solutions of Painlevé IV, arXiv:1905.10046.
  2. Dereziński J., Hypergeometric type functions and their symmetries, Ann. Henri Poincaré 15 (2014), 1569-1653, arXiv:1305.3113.
  3. Dereziński J., Group-theoretical origin of symmetries of hypergeometric class equations and functions, in Complex Differential and Difference Equations (Proceedings of the School and Conference Held at Bedlewo, Poland, September 2-15, 2018), Editors G. Filipuk, A. Lastra, S. Michalik, Y. Takei, H. Żołądek, De Gruyter Proceedings in Mathematics, Berlin, 2020, 3-128, arXiv:1906.03512.
  4. Dereziński J., Wrochna M., Exactly solvable Schrödinger operators, Ann. Henri Poincaré 12 (2011), 397-418, arXiv:1009.0541.
  5. Filipuk G., Ishkhanyan A., Dereziński J., On the derivatives of the Heun functions, J. Contemp. Math. Anal. 55 (2020), 200-207, arXiv:1907.12692.
  6. Fuchs R., Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen, Math. Ann. 63 (1907), 301-321.
  7. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes, Acta Math. 33 (1910), 1-55.
  8. Heun K., Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten, Math. Ann. 33 (1888), 161-179.
  9. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1944.
  10. Ishkhanyan A., Suominen K.-A., New solutions of Heun's general equation, J. Phys. A: Math. Gen. 36 (2003), L81-L85, arXiv:0909.1684.
  11. Its A.R., Prokhorov A., On some Hamiltonian properties of the isomonodromic tau functions, in Ludwig Faddeev Memorial Volume, World Sci. Publ., Hackensack, NJ, 2018, 227-264, arXiv:1803.04212.
  12. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  13. Jimbo M., Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
  14. Jimbo M., Miwa T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), 407-448.
  15. Kimura T., On Fuchsian differential equations reducible to hypergeometric equations by linear transformations, Funkcial. Ekvac. 13 (1971), 213-232.
  16. Maier R.S., The 192 solutions of the Heun equation, Math. Comp. 76 (2007), 811-843, arXiv:math.CA/0408317.
  17. Mason P., Differential equations and singularities II, available at
  18. Nikiforov A.F., Uvarov V.B., Special functions of mathematical physics. A unified introduction with applications, Birkhäuser Verlag, Basel, 1988.
  19. Ohyama Y., Okumura S., A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A: Math. Gen. 39 (2006), 12129-12151, arXiv:math.CA/0601614.
  20. Okamoto K., Isomonodromic deformation and Painlevé equations, and the Garnier system, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33 (1986), 575-618.
  21. Painlevé P., Mémoire sur les équations différentielles dont l'intégrale générale est uniforme, Bull. Soc. Math. France 28 (1900), 201-261.
  22. Painlevé P., Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), 1-85.
  23. Poole E.G.C., Linear differential equations, Oxford University Press, Oxford, 1936.
  24. Ronveaux A. (Editor), Heun's differential equations, Oxford University Press, Oxford, 1995.
  25. Slavyanov S.Yu., Lay W., Special functions. A unified theory based on singularities, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  26. Slavyanov S.Yu., Shatco D.A., Ishkhanyan A.M., Rotinyan T.A., Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients, Theoret. and Math. Phys. 189 (2016), 1726-1733, arXiv:1606.01476.

Previous article  Next article  Contents of Volume 17 (2021)