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Transactions of the American Mathematical Society

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Special values of multiple polylogarithms

Authors: Jonathan M. Borwein, David M. Bradley, David J. Broadhurst and Petr Lisonek
Journal: Trans. Amer. Math. Soc. 353 (2001), 907-941
MSC (2000): Primary 40B05, 33E20; Secondary 11M99, 11Y99
Published electronically: October 11, 2000
MathSciNet review: 1709772
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Abstract | References | Similar Articles | Additional Information


Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.

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Additional Information

Jonathan M. Borwein
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

David M. Bradley
Affiliation: Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, Maine 04469–5752

David J. Broadhurst
Affiliation: Physics Department, Open University, Milton Keynes, MK7 6AA, United Kingdom

Petr Lisonek
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Keywords: Euler sums, Zagier sums, multiple zeta values, polylogarithms, multiple harmonic series, quantum field theory, knot theory, Riemann zeta function.
Received by editor(s): July 29, 1998
Received by editor(s) in revised form: August 14, 1999
Published electronically: October 11, 2000
Additional Notes: The research of the first author was supported by NSERC and the Shrum Endowment of Simon Fraser University.
Article copyright: © Copyright 2000 American Mathematical Society