Convergence regions for the general continued fraction

Our result for /-fractions is stated in Theorem C. The results of this note are closely related to an as yet unpublished work of Wall and Wetzel on "positive definite /-fractions." In particular Theorem C seems to be contained in a theorem of theirs. In what follows we shall denote by H(b, y) the half-plane (including the boundary) defined by the relation z(E.H{b, 7) if dt(ze~) ^b. For the open half-plane we shall use the notation H^b, 7) . I t is clear from the context that b is a real number. Further for a >0 , P(a, 7) shall be the parabolic region (including the boundary) bounded by the curve a/2 P ^ 1 cos (6 27) For a = 0, P(a, 7) is to be the totality of points re, r ^ O .