Special values of multiple polylogarithms |
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Authors: | Jonathan M Borwein David M Bradley David J Broadhurst Petr Lisonek |
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Institution: | Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 ; Department of Mathematics and Statistics, University of Maine, 5752 Neville Hall, Orono, Maine 04469--5752 ; Physics Department, Open University, Milton Keynes, MK7 6AA, United Kingdom ; Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 |
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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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Keywords: | Euler sums Zagier sums multiple zeta values polylogarithms multiple harmonic series quantum field theory knot theory Riemann zeta function |
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