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

For a non-negative integer $N$, let $\mathcal{Z}_{N}:=\sum^N_{c = 0} \mathbb{Q} \cdot ζ(-c,s+c)$, 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 $\mathcal{Z}_{N} =\bigoplus^{N}_{c = 0 : \text{even}} \mathbb{Q} \cdot ζ(-c,s+c)$, where $\bigoplus$ is the direct sum of vector spaces. Moreover, we give a family of relations that exhaust all $\mathbb{Q}$-linear relations on $\mathcal{Z}_{N}$.

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