Iterated integrals on ${\mathbb{P}}^1\setminus \{0,1,\infty ,z\}$ and a class of relations among multiple zeta values

Abstract

In this paper we consider iterated integrals on ${\mathbb{P}}^1\setminus \{0,1,\infty ,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to $z$. We then define a new class of $\mathbb{Q}$-linear relations among the multiple zeta values by taking their limits of $z\to 1$, which we call \emph{confluence relations} (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values. As a good reason for this, we show that confluence relations imply both the regularized double shuffle relations and the duality relations.