Realizations of the double shuffle relations

Abstract

The set of multiple zeta values satisfy the double shuffle relations. In this paper, we study potential realizations of the double shuffle relations from multiple series and iterated path integrals. We show that the realizations of the double shuffle relations which can be constructed from multiple series are unique up to multiplying by a constant. We prove that there are no nontrivial realizations of the double shuffle relations from iterated path integrals of continuous functions on closed intervals. Lastly we show that the set of one variable multiple polylogarithms satisfies the double shuffle relations only for the point 0 and 1.