Partial sums of hypergeometric series of unit argument
The asymptotic behaviour of partial sums of generalized hypergeometric series of unit argument is investigated.
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arXiv subjects
Wolfgang Buehring
Publications and source records attributed to Wolfgang Buehring.
Analytic continuation of the generalized hypergeometric series near unit argument with emphasis on the zero-balanced series
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.
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An asymptotic expansion for a ratio of products of gamma functions
An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.
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Second-order linear differential equations with two irregular singular points of rank three: the characteristic exponent
For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the characteristic exponent of the multiplicative solutions. The Stokes multipliers are represented by converging series with terms for which limit formulas as well as more detailed asymptotic expansions are available. Here certain new, recursively known coefficients enter, which are closely related to but different from the coefficients of the formal solutions at one of the irregular singular points of the differential equation. The coefficients of the formal solutions then appear as finite sums over subsets of the new coefficients. As a by-product, the leading exponential terms of the asymptotic behaviour of the late coefficients of the formal solutions are given, and this is a concrete example of the structural results obtained by Immink in a more general setting. The formulas displayed in this paper are not of merely theoretical interest, but they also are complete in the sense that they could be (and have been) implemented for computing accurate numerical values of the characteristic exponent, although the computational load is not small and increases with the rank of the singular point under consideration.
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