Sum formulas and duality theorems of multiple zeta values

Abstract

Multiple zeta values or r-ford Euler sums are defined byζ(α1,α2,…,αr)=∑1≤n1<n2<⋯<nrn1−α1n2−α2⋯nr−αr with α1,α2,…,αr are positive integers and αr≥2. Let |α|=α1+α2+⋯+αr, the sum formula then asserted that∑|α|=mζ(α1,α2,…,αr+1)=ζ(m+1) for any positive integer m≥r.In this paper, we investigate the multiple zeta values with a parameter a>−1 given by∑1≤n1<n2<⋯<np+r1n1n2⋯np(np+1+a)⋯(np+r+a)np+r as well as their vector versions. Among other things, we obtain that the above multiple zeta values are equal to∑|α|=p+r∑1≤k1<k2<⋯<kp+1k1−α1k2−α2⋯kp+1−αp+1(kp+1+a)−1. By differentiations with respect to a and then set a=0, we obtain some interesting sum formulas including the aforementioned sum formula and the restricted sum formula∑|α|=mζ({1}p,α1,…,αr+1)=∑|c|=p+rζ(c1,c2,…,cp+1+(m−r)+1). Also, a generalized duality theorem is given in the final section.