Algorithms for Some Euler-Type Identities for Multiple Zeta Values

Abstract

Multiple zeta values are the numbers defined by the convergent series $\zeta (s_1,s_2,\dots ,s_k)={\sum }_{n_1>n_2>\cdots >n_k>0}(1/n_1^{s_1}\hspace{0.17em}{n}_2^{s_2}\cdots n_k^{s_k})$, where $s_1$, $s_2$, $\dots$, $s_k$ are positive integers with $s_1>1$. For $k\le n$, 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 $E(2n,2)=3\zeta (2n)/4$ 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 $k\le n$. 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.