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 $ζ\left(\overline{r}, s\right)$, $ζ\left(r, \overline{s}\right)$ and $ζ\left(\overline{r}, \overline{s}\right)$ 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 $ζ(2,\ldots,2,3,2,\ldots,2)$.