Appell Hypergeometric Expansions of the Solutions of the General Heun Equation



Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.


Linear ordinary differential equation Heun equation Special functions Series expansions Recurrence relations 

Mathematics Subject Classification

33E30 34B30 30Bxx 



I thank the referee for bringing to my attention the important contribution by Iwasaki et al. as well as for the important observation concerning the “desingularization” through a third-order ODE satisfied by v(z) and, as a result, possible three-term reduction of the recurrence relations for the coefficients of the series derived here. I am grateful to Professor Peter Olver for his careful reading of the manuscript and helpful suggestions. This research has been conducted within the scope of the International Associated Laboratory (CNRS-France & SCS-Armenia) IRMAS. The work has been supported by the State Committee of Science of the Republic of Armenia (project 18RF-139), the Armenian National Science and Education Fund (ANSEF Grant PS-4986), and the project “Leading Research Universities of Russia” (Grant FTI-24 -2016 of the Tomsk Polytechnic University).


  1. 1.
    Appell, P., Kampe de Feriet, M.J.: Fonctions hypergéométriques et hypersphériques: polynomes d’Hermite. Gauthier-Villars, Paris (1926)MATHGoogle Scholar
  2. 2.
    Erdélyi, A.: Certain expansions of solutions of the Heun equation. Q. J. Math. (Oxford) 15, 62 (1944)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Heun, K.: Zur Theorie der Riemann’schen Functionen zweiter Ordnung mit vier Verzweigungspunkten. Math. Ann. 33, 161 (1889)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ishkhanyan, A.M., Suominen, K.-A.: New solutions of Heun’s general equation. J. Phys. A 36, L81 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Iwasaki, K., Kimura, H., Shimomura, S., Yoshida, M.: From Gauss to Painlevé: A Modern Theory of Special Functions, Aspects of Mathematics, vol. 16. Vieweg, Braunschweig (1991)MATHGoogle Scholar
  6. 6.
    Kuiken, K.: Heun’s equation and the hypergeometric equation. SIAM J. Math. Anal. 10, 655 (1979)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Leroy, C., Ishkhanyan, A.M.: Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions. Integral Transforms Spec. Funct. 26, 451 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maier, R.S.: On reducing the Heun equation to the hypergeometric equation. J. Differ. Equ. 213, 171 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966)MATHGoogle Scholar
  10. 10.
    Ronveaux, A. (ed.): Heun’s Differential Equations. Oxford University Press, London (1995)MATHGoogle Scholar
  11. 11.
    Slavyanov, S.Yu., Lay, W.: Special Functions. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  12. 12.
    Schmidt, D.: Die Lösung der linearen Differentialgleichung 2. Ordnung um zwei einfache Singularitäten durch Reihen nach hypergeometrischen Funktionen. J. Reine Angew. Math. 309, 127 (1979)MathSciNetMATHGoogle Scholar
  13. 13.
    Svartholm, N.: Die Lösung der Fuchs’schen Differentialgleichung zweiter Ordnung durch Hypergeometrische Polynome. Math. Ann. 116, 413 (1939)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Vidunas, R., Filipuk, G.: Parametric transformations between the Heun and Gauss hypergeometric functions. Funkcialaj Ekvacioj 56, 271 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wimp, J.: Computation with Recurrence Relations. Pitman, London (1983)MATHGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Physical Research of NAS of ArmeniaAshtarakArmenia
  2. 2.Institute of Physics and TechnologyNational Research Tomsk Polytechnic UniversityTomskRussia

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