Alternating Double Euler Sums, Hypergeometric Identities and a Theorem of Zagier

Abstract

In this work, we derive relations between generating functions of double stuffle relations and double shuffle relations to express the alternating double Euler sums $\zeta \left(\overline{r},s\right)$, $\zeta \left(r,\overline{s}\right)$ and $\zeta \left(\overline{r},\overline{s}\right)$ with $r+s$ odd in terms of zeta values. We also give a direct proof of a hypergeometric identity which is a limiting case of a basic hypergeometric identity of Andrews. Finally, we gave another proof for the formula of Zagier on the multiple zeta values $\zeta (2,\dots ,2,3,2,\dots ,2)$.

25 pages