Difference between revisions of "Q-factorial"

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The $q$-Factorial is defined for a non-negative integer $k$ by
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__NOTOC__
$$[k]_q!=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{k-1})=\dfrac{(q;q)_k}{(1-q)^k},$$
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The $q$-factorial is defined for a non-negative integer $k$ by
where $(q;q)_k$ is the [[q-Pochhammer symbol]].
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$$[n]_q! = \displaystyle\prod_{k=1}^n [k]_q= \left( \dfrac{1-q}{1-q} \right) \left( \dfrac{1-q^2}{1-q} \right) \ldots \left( \dfrac{1-q^n}{1-q} \right),$$
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where $[k]_q$ denotes a [[q-number|$q$-number]].
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=Properties=
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[[Q-derivative power rule]]<br />
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=See Also=
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[[q-number|$q$-number]]<br />
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[[q-Pochhammer|$q$-Pochhammer]]<br />
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=References=
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* {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=Q-exponential E sub q|next=findme}}
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* {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=q-number|next=findme}}
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* {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(3.1)$
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* {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-number when a=n is a natural number|next=findme}}: ($6.3$)
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{{:q-calculus footer}}
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[[Category:SpecialFunction]]

Latest revision as of 04:23, 26 December 2016

The $q$-factorial is defined for a non-negative integer $k$ by $$[n]_q! = \displaystyle\prod_{k=1}^n [k]_q= \left( \dfrac{1-q}{1-q} \right) \left( \dfrac{1-q^2}{1-q} \right) \ldots \left( \dfrac{1-q^n}{1-q} \right),$$ where $[k]_q$ denotes a $q$-number.

Properties

Q-derivative power rule

See Also

$q$-number
$q$-Pochhammer

References

$q$-calculus