Double weighted sum formulas of multiple zeta values

Abstract

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

$$\begin{aligned} \sum _{1 \le n_{1} < n_{2} < \cdots < n_{r}} n_{1}^{-\alpha _{1}} n_{2}^{-\alpha _{2}} \ldots n_{r}^{-\alpha _{r}}. \end{aligned}$$

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.