On Multiple Zeta Values of Even Arguments

TL;DR

This paper derives formulas for sums of multiple zeta values with even arguments across various depths, generalizing previous results and establishing new identities involving Bernoulli numbers.

Contribution

It provides two formulas for E(2n,k) valid for all k <= n, one generalizing earlier work, and introduces generating functions for these sums and their star value counterparts.

Findings

Two formulas for E(2n,k) valid for all k <= n

A Bernoulli-number identity derived from formula comparison

Explicit generating functions for E(2n,k) and E*(2n,k)

Abstract

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. Of course E(2n,1) is the value of the Riemann zeta function at 2n, and it is well known that E(2n,2) = (3/4)E(2n,1). Recently Z. Shen and T. Cai gave formulas for E(2n,3) and E(2n,4). We give two formulas form E(2n,k), both valid for arbitrary k <=n, one of which generalizes the Shen-Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers E(2n,k) and for the analogous numbers E*(2n,k) defined using multiple zeta-star values of even arguments.