  We examine the expansions of the solutions of the general Heun equation in
terms of the Gauss hypergeometric functions. We present several expansions
using functions, the forms of which differ from those applied before. In
general, the coefficients of the expansions obey three-term recurrence
relations. However, there exist certain choices of the parameters for which the
recurrence relations become two-term. The coefficients of the expansions are
then explicitly expressed in terms of the gamma functions. Discussing the
termination of the presented series, we show that the finite-sum solutions of
the general Heun equation in terms of generally irreducible hypergeometric
functions have a representation through a single generalized hypergeometric
function. Consequently, the power-series expansion of the Heun function for any
such case is governed by a two-term recurrence relation.
