Weighted Sum Formulas from Shuffle Products of Multiple Zeta-star Values

TL;DR

This paper derives a weighted sum formula involving shuffle products of multiple zeta-star values, revealing new relations and formulas for these special functions.

Contribution

It introduces a novel weighted sum formula from shuffle products of specific multiple zeta values and zeta-star values, expanding the understanding of their algebraic relations.

Findings

Derived a new weighted sum formula involving multiple zeta values.

Established explicit relations between shuffle products and weighted sums.

Provided a formula connecting zeta functions with combinatorial weights.

Abstract

In this paper, we are going to perform the shuffle products of $Z_-(n) = \sum_{a+b=m} (-1)^{b} \zeta(\{1\}^{a},b+2)$ and $Z_+^\star(n) = \sum_{c+d=n} \zeta^{\star}(\{1\}^{c},d+2)$ with $m+n = p$. The resulted shuffle relation is a weighted sum formula given by \begin{equation*} \frac{(p+1)(p+2)}{2} \zeta(p+4) =\sum_{m+n=p} \sum_{|\boldsymbol{\alpha}|=p+3} \zeta(\alpha_{0}, \alpha_{1}, \ldots, \alpha_{m}, \alpha_{m+1}+1) \sum_{a+b+c=m} \Bigl( W_{\boldsymbol\alpha}(a,b,c) + W_{\boldsymbol\alpha}(a,b,c=0) + W_{\boldsymbol\alpha}(a=0,b,c) + W_{\boldsymbol\alpha}(a=0,b=m,c=0) \Bigr), \end{equation*} where $W_{\boldsymbol\alpha}(a,b,c) = 2^{\sigma(a+b+1)-\sigma(a)-(b+1)} (1-2^{1-\alpha_{a+b+1}}\ \ )$, with $\sigma(r) = \sum_{j=0}^{r} \alpha_{j}$.