Additive polylogarithms and their functional equations

Let ${k[varepsilon]_{2}:=k[varepsilon]/(varepsilon^{2})}Let k[e]₂:=k[e]/(e²){k[varepsilon]_{2}:=k[varepsilon]/(varepsilon^{2})} . The single valued real analytic n-polylogarithm L_n: mathbbC ? mathbbR{mathcal{L}_{n}: mathbb{C} to mathbb{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 ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms li_n:k[e]₂? k{li_{n}:k[varepsilon]_{2}to k} and prove that they satisfy functional equations analogous to those of L_n{mathcal{L}_{n}}. Under a mild hypothesis, we show that these functions descend to an analog of the nth Bloch group B_n￠(k[e]₂){B_{n}' (k[varepsilon]_{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[ε]₂.