Appell Hypergeometric Expansions of the Solutions of the General Heun Equation

A. M. Ishkhanyan

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)445-459
Number of pages15
JournalConstructive Approximation
Issue number3
Publication statusPublished - 15 Jun 2019


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

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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