Recursion Formulae for Hypergeometric Functions
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چکیده
indicates the term corresponding to j = h is to be deleted. If one of the a, = 0 or a negative integer, then (1) always converges, since it terminates. Otherwise it converges for all finite x if P ^ Q and for |a;| < 1 if P = Q + 1. In this case, however, the function can be analytically continued into the cut plane |arg (1 — a;)| < x, and we shall often denote by q+iFq(x) not only the series (1), whenever it converges, but also the analytic continuation of the series. If P > Q + 1, the series does not converge (unless it terminates) and if one of the b¡ is 0 or a negative integer, the series is not defined. If one of the at equals one of the bj, pFQ(x) reduces to P-iFQ-i(x) and such a case is always excluded from consideration in this paper. We assume all pFQ's are irreducible. Equation (1) can be given an interpretation for P > Q + 1 by means of the G-function
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