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For a zero-balanced generalized hypergeometric function „F x(z), the authors prove a formula exhibiting its behavior near the boundary point z = 1 of the region of convergence of the series defining it. The result established here provides an interesting extension of a formula which appeared in one of Ramanujan's celebrated Notebooks; it also serves to solve the problem posed by R. J. Evans [5]. In many areas of applications involving various hypergeometric functions of one, two, and more variables, one finds the need to know the behaviors of these functions near the boundaries of the regions of convergence of the series defining them. Several results of this type have appeared in the literature (cf., e.g., [11]-[16]); some of these results were used in solving various boundary value problems involving the Euler-Darboux equation and in the study of certain operators of fractional calculus (cf. [10] and [18]). The behavior of the Gaussian hypergeometric function 2Fx(z) near z = 1 is given by a well-known analytic continuation formula [4, p. 108, Equation 2.10(1)]. An asymptotic formula exhibiting the behavior of a zero-balanced Clausenian hypergeometric function 3F2(z) near z = 1 was stated (without proof) by Ramanujan ([9, Chapter 11, Entry 24, Corollary 2]; see also [2, p. 295, Corollary 2]). A generalized hypergeometric function ,7' (z) defined, in terms of the Pochhammer symbol (X)n T(X + n)/T(X), by (D p+iFp[(ap+x);(bp);z] = p+xFp Received by the editors January 4, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 33A30; Secondary 35F15.

The behavior of the zero-balanced hypergeometric series ${}\sb pF\sb {p-1}$ near the boundary of its convergence region