Some variations of multiple zeta values

Abstract

In this paper, we build some variations of multiple zeta values and investigate their relations. Among other things, we prove that \[ \sum _{\substack {|\boldsymbol {\alpha }|=m+r\\ 1\leq k_1\lt k_2\lt \cdots \lt k_r}} \!\!\!\!\!\!\!\!k_1^{-\alpha _1}k_2^{-\alpha _2} \cdots k_r^{-\alpha _r}(k_r-p)^{-1} \] can be evaluated as a linear combination of $\zeta (r), \zeta (r-1), \ldots , \zeta (r-p+1)$ for $r\geq p+1$. In particular, for $r\geq 2$, \[ \sum _{\substack {|{\bf \alpha }|=m+r\\ 1\leq k_1\lt k_2\lt \cdots \lt k_r}} \!\!\!\!\!\!\!\!k_1^{-\alpha _1}k_2^{-\alpha _2} \cdots k_r^{-\alpha _r}(k_r-1)^{-1}=\zeta (r), \] which may be compared to the well-known sum formula. A similar discussion leads to the twisted sum formula.