  Starting from the equation obeyed by the derivative, we construct several
expansions of the solutions of the general Heun equation in terms of the Appell
generalized hypergeometric functions of two variables of the fist kind. Several
cases when the expansions reduce to ones written in terms of simpler
mathematical functions such as the incomplete Beta function or the Gauss
hypergeometric function are identified. The conditions for deriving finite-sum
solutions via termination of the series are discussed. In general, the
coefficients of the expansions obey four-term recurrence relations; however,
there exist certain sets of the parameters for which the recurrence relations
involve only two terms, though not successive. The coefficients of the
expansions are then explicitly calculated and the general solution of the Heun
equation is constructed in terms of the Gauss hypergeometric functions.
