Computation and theory of extended Mordell-Tornheim-Witten sums

Bailey, David; Borwein, Jonathan; Crandall, Richard

doi:10.1090/S0025-5718-2014-02768-3

Computation and theory of extended Mordell-Tornheim-Witten sums
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by David H. Bailey, Jonathan M. Borwein and Richard E. Crandall;

Math. Comp. 83 (2014), 1795-1821

DOI: https://doi.org/10.1090/S0025-5718-2014-02768-3

Published electronically: January 23, 2014

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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 an 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.

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