论文标题
来自多个Zeta-Star值的洗牌产品的加权总和公式
Weighted Sum Formulas from Shuffle Products of Multiple Zeta-star Values
论文作者
论文摘要
在本文中，我们将执行$ z _-（n）= \ sum_ {a+b = m}（-1）^{b} pary的混洗产品ζ^{\ star}（\ {1 \}^{c}，d+2）$，$ m+n = p $。所得的洗牌关系是\ begin {equation*}给出的加权总和公式 \ 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 {qore {qore*} $ w _ {\boldsymbolα}（a，b，b，c）（a，b，c） （1-2^{1-α_{a+b+1}}} \ \）$，带有$σ（r）= \ sum_ {j = 0}^{r}α__{j} $。
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 $σ(r) = \sum_{j=0}^{r} α_{j}$.
