Weighted and Restricted Sum Formulas of Euler Sums

Abstract

One of the most interesting formulas for multiple zeta values is the sum formula proved by Granville and Zagier independently in 1990s. Many variations and generalizations of it have been found since then. In this paper, we will provide a uniform approach to proving these results using the regularized double shuffle relations of Euler sums. We summarize these formulas including many new ones with three tables in the Appendix at the end of the paper.