  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.
