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

Abstract

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