Difference between revisions of "Q-shifted factorial"

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The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$, by
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The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$ or $n=\infty$, by
$$\displaystyle\prod_{k=0}^{n-1} 1-aq^{k-1}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}).$$
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$$(a;q)_n=\left\{
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\begin{array}{ll}
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1, & \quad n=0 \\
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\displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}), & \quad n=1,2,3,\ldots \\
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\end{array} \right.$$
  
 
=Properties=
 
=Properties=
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 +
=See Also=
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[[Q-Pochhammer]]<br />
  
 
=References=
 
=References=
* {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=findme|next=findme}} $(10.2.1)$ (does not specifically say "$q$-shifted factorial")
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* {{BookReference|Higher Transcendental Functions Volume I|1953|Arthur Erdélyi|author2=Wilhelm Magnus|author3=Fritz Oberhettinger|author4=Francesco G. Tricomi|prev=findme|next=Basic hypergeometric phi}}: $4.8 (1) (2)$ (assumes $|q|<1$)
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* {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-Gamma|next=Q-Gamma at z+1}}
 
* {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}}  
 
* {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}}  
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* {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=findme|next=findme}} $(10.2.1)$
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 01:14, 25 December 2018

The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$ or $n=\infty$, by $$(a;q)_n=\left\{ \begin{array}{ll} 1, & \quad n=0 \\ \displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}), & \quad n=1,2,3,\ldots \\ \end{array} \right.$$

Properties

See Also

Q-Pochhammer

References