Difference between revisions of "Q-zeta"
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| − | The $q$-zeta function $\zeta_q \colon \mathbb{C} \times (0,1] \rightarrow \mathbb{C}$ is defined by | + | 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 |
$$\zeta_q(z,x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k]+x)^z},$$ | $$\zeta_q(z,x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k]+x)^z},$$ | ||
where $[k]$ denotes a [[q-number|$q$-number]]. | where $[k]$ denotes a [[q-number|$q$-number]]. | ||
Revision as of 17:46, 11 February 2018
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 $$\zeta_q(z,x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k]+x)^z},$$ where $[k]$ denotes a $q$-number.
