Special values of multiple polylogarithms

Borwein, Jonathan; Bradley, David; Broadhurst, David; Lisoněk, Petr

doi:10.1090/S0002-9947-00-02616-7

Special values of multiple polylogarithms
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by Jonathan M. Borwein, David M. Bradley, David J. Broadhurst and Petr Lisoněk

Trans. Amer. Math. Soc. 353 (2001), 907-941

DOI: https://doi.org/10.1090/S0002-9947-00-02616-7

Published electronically: October 11, 2000

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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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