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We examine the expansions of the solutions of the general Heun equation interms of the Gauss hypergeometric functions. We present several expansionsusing functions, the forms of which differ from those applied before. Ingeneral, the coefficients of the expansions obey three-term recurrencerelations. However, there exist certain choices of the parameters for which therecurrence relations become two-term. The coefficients of the expansions arethen explicitly expressed in terms of the gamma functions. Discussing thetermination of the presented series, we show that the finite-sum solutions ofthe general Heun equation in terms of generally irreducible hypergeometricfunctions have a representation through a single generalized hypergeometricfunction. Consequently, the power-series expansion of the Heun function for anysuch case is governed by a two-term recurrence relation.

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