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The Appell hypergeometric expansions of the solutions of the general Heun equation

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.