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

Abstract

In this paper, we are going to perform the shuffle products of $$Z_-({m}) = \sum _{a+b=m} (-1)^{b} \zeta (\{1\}^{a},b+2)$$ 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 stated below: $$\begin{aligned}&\frac{(p+1)(p+2)}{2} \zeta (p+4) =\sum _{\begin{array}{c} m+n=p\\ \mid \varvec{\alpha }\mid =p+3 \end{array}} \zeta (\alpha _{0}, \alpha _{1}, \ldots , \alpha _{m}, \alpha _{m+1}+1) \\&\times {\sum _{a+b+c=m}}^{*} \Bigl ( W_{\varvec{\alpha }}(a,b,c) + W_{\varvec{\alpha }}(a,b,0) + W_{\varvec{\alpha }}(0,b,c)+ W_{\varvec{\alpha }}(0,m,0) \Bigr ), \end{aligned}$$ where the summation $$\sum ^*$$ and the weights $$W_{\varvec{\alpha }}(a,b,c)$$ are given appropriately. We also give some weighted alternating Euler sums formulas.