The Appell hypergeometric expansions of the solutions of the general Heun equation

Abstract

Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases when the expansions reduce to ones 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 sets of the parameters for which the recurrence relations involve only two terms, though not successive. The coefficients of the expansions are then explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

References in corpus (8)

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• Heun Functions and Some of Their Applications in Physics
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Cited by in corpus (5)

• Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions
• Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation
• Physical Problems Admitting Heun-to-Hypergeometric Reduction
• Expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions
• Generalized-hypergeometric solutions of the biconfluent Heun equation