Alternating double Euler sums, hypergeometric identities and a theorem of Zagier

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(r‾,s), ζ(r,s‾) and ζ(r‾,s‾) 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,…,2,3,2,…,2).