Double Shuffle Relations of Euler Sums

Jianqiang Zhao

Published 2007-05-16, updated 2007-07-23Version 5

In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, \emph{Derivation and double shuffle relations for multiple zeta values}. Compos. Math. \textbf{142} (2)(2006), 307--338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the end we shall prove the following long standing conjecture: for every positive integer n $$\zeta(\{3\}^n)=8^n\zeta(\{\ol2,1\}^n).$$ The main idea is to use the double shuffle relations and the distribution relation. This particular distribution relation doesn't follow from the double shuffle relation in general. But we believe it does follow from the extended double shuffle relations.