Q-zeta

Let $|q|<1$. The $q$-zeta function $\zeta_q \colon \mathbb{C} \times (0,1] \rightarrow \mathbb{C}$ is defined for $\mathrm{Re}(z)>1$ by $$\zeta_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k])^z},$$ where $[k]$ denotes a $q$-number.