On a basis for Euler-Zagier double zeta functions with non-positive components

Abstract: For a non-negative integer $N$, let $\mathcal{Z}{N}:=\sum^N{c = 0} \mathbb{Q} \cdot \zeta(-c,s+c)$, where the right-hand side is the vector space spanned by the Euler-Zagier double zeta functions over $\mathbb{Q}$. In this paper, we show that $\mathcal{Z}{N} =\bigoplus^{N}{c = 0 : \text{even}} \mathbb{Q} \cdot \zeta(-c,s+c)$, where $\bigoplus$ is the direct sum of vector spaces. Moreover, we give a family of relations that exhaust all $\mathbb{Q}$-linear relations on $\mathcal{Z}_{N}$.