Special Values of Multiple Polylogarithms
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, Petr Lisonek
TL;DR
The paper builds a unified framework for special values of multiple polylogarithms that bridges classical polylogarithms, Euler sums, and multiple zeta values through the $\lambda$-notation. It develops robust integral representations (partition and iterated forms), shuffle/stuffle algebras, and duality to derive broad reductions and prove key evaluations, including Zagier's conjecture, while offering computational tools like the Hölder convolution and EZ Face for high-precision computation. It also connects functional equations and differential equations to derive new identities and generalizations, revealing deep algebraic and analytic structures behind MZVs. Collectively, the work advances both theoretical understanding and practical computation of a wide class of polylogarithmic sums with significant implications in number theory, combinatorics, and mathematical physics.
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.