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Applying the approach based on the equation for the derivative, we constructseveral expansions of the solutions of the general Heun equation in terms ofthe incomplete Beta functions. Several expansions in terms of the Appellgeneralized hypergeometric functions of two variables of the fist kind are alsopresented. The constructed expansions are applicable for arbitrary sets of theinvolved parameters. The coefficients of the expansions obey four-, five- orsix-term recurrence relations. However, there exist several sets of theparameters for which the recurrence relations involve fewer terms, notnecessarily successive. The conditions for deriving finite-sum solutions viatermination of the series are discussed.

Expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions