Algorithms for some euler-type identities for multiple zeta values

Multiple zeta values are the numbers defined by the convergent series ζ (s 1,s 2,.,s k) = + n 1 > n 2 > > n k > 0 (1 / n 1 s 1 n 2 s 2 n k s k),where s 1,s 2,.,s k are positive integers with s 1 > 1. for k ≤ n,let e (2 n,k) be the sum of all multiple zeta values with even arguments whose weight is 2 n and whose depth is k. the well-known result e (2 n,2) = 3 ζ (2 n) / 4 was extended to e (2 n,3) and e (2 n,4) by z. shen and t. cai. applying the theory of symmetric functions,hoffman gave an explicit generating function for the numbers e (2 n,k) and then gave a direct formula for e (2 n,k) for arbitrary k ≤ n. in this paper we apply a technique introduced by granville to present an algorithm to calculate e (2 n,k) and prove that the direct formula can also be deduced from eisenstein's double product. © 2013 shifeng ding and weijun liu.