On the convolutions of sums of multiple zeta(-star) values of height one

TL;DR

This paper explores the properties of sums of multiple zeta(-star) values of height one, providing new evaluations of complex weighted sums through convolution techniques.

Contribution

It introduces a novel approach to evaluate sums of multiple zeta(-star) values of height one using convolution of specific sums, advancing understanding in this area.

Findings

Weighted sums of multiple zeta(-star) values can be expressed via convolution.

New explicit evaluations for sums involving height-one multiple zeta values.

The convolution approach simplifies complex sum evaluations.

Abstract

In this paper, we investigate the sums of mutliple zeta(-star) values of height one: $Z_{\pm}(n)=\sum_{a+b=n} (\pm 1)^b\zeta(\{1\}^a,b+2)$, $Z_{\pm}^{\star}(n)=\sum_{a+b=n} (\pm 1)^b\zeta^{\star}(\{1\}^a,b+2)$. In particular, we prove that the weighted sum $\sum_{\substack{0\leq m\leq p\\ m: {\rm even}}} \sum_{\mid\boldsymbol{\alpha}\mid=p+3} 2^{\alpha_{m+1}\ +1}\zeta(\alpha_0,\alpha_1,\ldots,\alpha_m,\alpha_{m+1}+1) $ can be evaluated through the convolution of $Z_{-}(m)$ and $Z_{+}(n)$ with $m+n=p$.