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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Special values of multiple polylogarithms
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by Jonathan M. Borwein, David M. Bradley, David J. Broadhurst and Petr Lisoněk PDF
Trans. Amer. Math. Soc. 353 (2001), 907-941 Request permission

Abstract:

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
  • Email: jborwein@cecm.sfu.ca
  • David M. Bradley
  • Affiliation: Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, Maine 04469–5752
  • MR Author ID: 329306
  • ORCID: 0000-0003-2952-2366
  • Email: bradley@gauss.umemat.maine.edu, dbradley@member.ams.org
  • David J. Broadhurst
  • Affiliation: Physics Department, Open University, Milton Keynes, MK7 6AA, United Kingdom
  • Email: D.Broadhurst@open.ac.uk
  • Petr Lisoněk
  • Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
  • Email: lisonek@cecm.sfu.ca
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 907-941
  • MSC (2000): Primary 40B05, 33E20; Secondary 11M99, 11Y99
  • DOI: https://doi.org/10.1090/S0002-9947-00-02616-7
  • MathSciNet review: 1709772