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

Applying the approach based on the equation for the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions. Several expansions in terms of the Appell generalized hypergeometric functions of two variables of the fist kind are also presented. The constructed expansions are applicable for arbitrary sets of the involved parameters. The coefficients of the expansions obey four-, five- or six-term recurrence relations. However, there exist several sets of the parameters for which the recurrence relations involve fewer terms, not necessarily successive. The conditions for deriving finite-sum solutions via termination of the series are discussed.