Special Values of Multiple Polylogarithms

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 Introduction We are going to study a class of multiply nested sums of the form s sk b bk X