Additive polylogarithms and their functional equations

Let k[ε]2 := k[ε]/(ε2). The single valued real analytic n-polylogarithm Ln : 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 Ünver (Algebra Number Theory 3:1–34, 2009) to define additive n-polylogarithms lin:k[ε]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[ε]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.