Analytic continuation of the generalized hypergeometric series near unit argument with emphasis on the zero-balanced series

Abstract

Various methods to obtain the analytic continuation near $z=1$ of the hypergeometric series ${}_{p+1}{F}_p(z)$ are reviewed together with some of the results. One approach is to establish a recurrence relation with respect to $p$ and then, after its repeated use, to resort to the well-known properties of the Gaussian hypergeometric series. Another approach is based on the properties of the underlying generalized hypergeometric differential equation: For the coefficients in the power series expansion around $z=1$ a general formula, valid for any $p$, is found in terms of a limit of a terminating Saalschützian hypergeometric series of unit argument. Final results may then be obtained for each particular $p$ after application of an appropriate transformation formula of the Saalschützian hypergeometric series. The behaviour at unit argument of zero-balanced hypergeometric series, which have received particular attention in recent years, is discussed in more detail. The related problem involving the behaviour of partial sums of such series is addressed briefly.

19 pages; v2: equations (4.10) and (4.11) corrected

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