Algorithms for Some Euler-Type Identities for Multiple Zeta Values

Abstract EN

Multiple zeta values are the numbers defined by the convergent series ζ(s1,s2,…,sk)=∑n1>n2>⋯>nk>0(1/n1s1 n2s2⋯nksk), where s1, s2, …, sk are positive integers with s1>1.

For k≤n, let E(2n,k) be the sum of all multiple zeta values with even arguments whose weight is 2n and whose depth is k.

The well-known result E(2n,2)=3ζ(2n)/4 was extended to E(2n,3) and E(2n,4) by Z.

Shen and T.

Cai.

Applying the theory of symmetric functions, Hoffman gave an explicit generating function for the numbers E(2n,k) and then gave a direct formula for E(2n,k) for arbitrary k≤n.

In this paper we apply a technique introduced by Granville to present an algorithm to calculate E(2n,k) and prove that the direct formula can also be deduced from Eisenstein's double product.