Appell Hypergeometric Expansions of the Solutions of the General Heun Equation

  • A. M. Ishkhanyan


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).


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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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