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
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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
$$\zeta_q(z,x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k]+x)^z},$$
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$$\zeta_q(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{q^{-k}}{(q^{-k}[k])^z},$$
 
where $[k]$ denotes a [[q-number|$q$-number]].
 
where $[k]$ denotes a [[q-number|$q$-number]].
  
 
=Properties=
 
=Properties=
  
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=See also=
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[[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}}  
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* {{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.

Properties

See also

$q$-Hurwitz zeta

References