Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients

Abstract

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

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Cited by in corpus (6)

• Non-commutativity and non-inertial effects on the Dirac oscillator in a cosmic string space-time
• Generalized-hypergeometric solutions of the general Fuchsian linear ODE having five regular singularities
• Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations
• A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations
• Solutions of Heun's general equation and elliptic Darboux equation
• The third five-parametric hypergeometric quantum-mechanical potential