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

Abstract

In this paper, we are going to perform the shuffle products of $Z_{-}(n)={\sum }_{a+b=m}(-1{)}^b\zeta (\{1{\}}^a,b+2)$ 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} 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 $W_{\mathbf{\alpha }}(a,b,c)=2^{\sigma (a+b+1)-\sigma (a)-(b+1)}(1-2^{1-{\alpha }_{a+b+1}})$, with $\sigma (r)={\sum }_{j=0}^r{\alpha }_j$.

17 pages