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 |
| − | $$\zeta_q(z | + | $$\zeta_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k])^z},$$ |
where $[k]$ denotes a [[q-number|$q$-number]]. | where $[k]$ denotes a [[q-number|$q$-number]]. | ||
=Properties= | =Properties= | ||
| + | =See also= | ||
| + | [[q-Hurwitz zeta|$q$-Hurwitz zeta]]<br /> | ||
=References= | =References= | ||
| − | * {{PaperReference|q-Dedekind type sums related to q-zeta function and basic L-series|2006|Yilmaz Simsek|prev=findme|next=findme}} | + | * {{PaperReference|q-Dedekind type sums related to q-zeta function and basic L-series|2006|Yilmaz Simsek|prev=findme|next=findme}}: $(2.4)$ |
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 04:51, 12 February 2018
Let $|q|<1$. The $q$-zeta function $\zeta_q \colon \mathbb{C} \times (0,1] \rightarrow \mathbb{C}$ is defined for $\mathrm{Re}(z)>1$ by $$\zeta_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k])^z},$$ where $[k]$ denotes a $q$-number.
