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Computation and Theory of Extended MordellL-Tornheim-Witten Sums

Bailey, David H. and Borwein, Jonathan M. and Crandall, Richard E. (2014) Computation and Theory of Extended MordellL-Tornheim-Witten Sums. Mathematics of Computation (83). pp. 1795-1821.

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    Abstract

    We consider some fundamental generalized Mordell-Tornheim-Witten (MTW) zeta-function values along with their derivatives, and explore connections with multiple- zeta values (MZVs). To achieve this, we make use of symbolic integration, high precision numerical integration, and some interesting combinatorics and special-function theory. Our original motivation was to represent unresolved constructs such as Eulerian log- gamma integrals. We are able to resolve all such integrals in terms of a MTW basis. We also present, for a substantial subset of MTW values, explicit closed-form expressions. In the process, we significantly extend methods for high-precision numerical computation of polylogarithms and their derivatives with respect to order.

    Item Type: Article
    Subjects: 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory
    33-xx Special functions > 33Bxx Elementary classical functions
    41-xx Approximations and expansions > 41Axx Approximations and expansions
    Faculty: UNSPECIFIED
    Depositing User: Jonathan Borwein
    Date Deposited: 10 Nov 2012 10:57
    Last Modified: 13 Jan 2015 12:30
    URI: https://docserver.carma.newcastle.edu.au/id/eprint/1459

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