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Computation and theory of extended Mordell-Tornheim-Witten sums


Authors: David H. Bailey, Jonathan M. Borwein and Richard E. Crandall
Journal: Math. Comp. 83 (2014), 1795-1821
MSC (2010): Primary 33B15, 33F05, 65D20, 65D30, 11M32
DOI: https://doi.org/10.1090/S0025-5718-2014-02768-3
Published electronically: January 23, 2014
MathSciNet review: 3194130
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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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Additional Information

David H. Bailey
Affiliation: Lawrence Berkeley National Laboratory, Berkeley, California 94720
Email: DHBailey@lbl.gov

Jonathan M. Borwein
Affiliation: CARMA, University of Newcastle, Callaghan, NSW 2308, Australia and Distinguished Professor King Abdulaziz University, Jeddah
Email: jonathan.borwein@newcastle.edu.au

Richard E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202
Email: crandall@reed.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02768-3
Received by editor(s): April 26, 2012
Received by editor(s) in revised form: August 1, 2012
Published electronically: January 23, 2014
Additional Notes: Richard Crandall passed away on December 20, 2012
Copyright Status: LBNL authored documents are sponsored by the U.S. Department of Energy under Contract DE-AC02-05CH11231. Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce these documents, or allow others to do so, for U.S. Government purposes. The documents may be freely distributed and used for noncommercial, scientific and educational purposes.

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