Double weighted sum formulas of multiple zeta values
详细信息    查看全文
文摘
For positives integers \(\alpha _{1}, \alpha _{2}, \ldots , \alpha _{r}\) with \(\alpha _{r} \ge 2\) , the multiple zeta value or \(r\) -fold Euler sum \(\zeta (\alpha _{1}, \alpha _{2}, \ldots , \alpha _{r})\) is defined by the multiple series $$\begin{aligned} \sum _{1 \le n_{1} In this paper, for integers \(k,r\ge 0\) and complex numbers \(\mu ,\lambda ,\) we consider the double weighted sum defined by $$\begin{aligned} E_{k,r}(\mu ,\lambda )=\sum _{p+q=k}\mu ^{p}\sum _{\left| \alpha \right| =q+r+3}\zeta ({\left\{ 1 \right\} ^{p},\alpha _{0},\ldots ,\alpha _{q},\alpha _{q+1}+1})\lambda ^{\alpha _{q+1}} \end{aligned}$$ and then evaluate \(E_{k,r}(2,2),\) \(E_{k,r}(2,1),\) \(E_{k,r}(1,2),\) \(E_{k,r}(1,1)\) and \(E_{k,r}(0,1)\) in terms of the special values at positive integers of the Riemann zeta function. Note that $$\begin{aligned} E_{k,r}(0,1)=\sum _{\left| \alpha \right| =k+r+3}\zeta (\alpha _{0},\ldots ,\alpha _{k},\alpha _{k+1}+1) \end{aligned}$$ so our results cover the sum formula $$\begin{aligned} \sum _{\left| \alpha \right| =k+r+3}\zeta (\alpha _{0},\ldots ,\alpha _{k},\alpha _{k+1}+1)=\zeta (k+r+4) \end{aligned}$$ proved by Granville in 1996.

© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号

地址：北京市海淀区学院路29号 邮编：100083

电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700