Additive polylogarithms and their functional equations

Abstract

Let k[epsilon](2) := k[epsilon]/(epsilon(2)). The single valued real analytic n-polylogarithm L-n : C -> R is fundamental in the study of weight n motivic cohomology over a field k, of characteristic 0. In this paper, we extend the construction in Unver (Algebra Number Theory 3:1-34, 2009) to define additive n-polylogarithms li(n):k[epsilon](2) -> k and prove that they satisfy functional equations analogous to those of Ln. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B'(n)(k[epsilon](2)) defined by Goncharov (Adv Math 114:197-318, 1995). We hope that these functions will be useful in the study of weight n motivic cohomology over k[epsilon](2).