  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 $\zeta(2,\ldots,2,3,2,\ldots,2)$.
