Realizations of the double shuffle relations

The set of multiple zeta values satisfy the double shuffie relations. In this paper, we study potential realizations of the double shuffie relations from multiple series and iterated path integrals. We show that the realizations of the double shuffie 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 shuffie 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 shuffie relations only for the point 0 and 1.& COPY; 2023 Elsevier Inc. All rights reserved.