Difference between revisions of "Q-factorial"
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| − | The $q$- | + | The $q$-factorial is defined for a non-negative integer $k$ by |
| − | $$[n]_q! = \displaystyle\prod_{k=1}^n [k]_q= | + | $$[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),$$ |
where $[k]_q$ denotes a [[q-number|$q$-number]]. | where $[k]_q$ denotes a [[q-number|$q$-number]]. | ||
=Properties= | =Properties= | ||
[[Q-derivative power rule]]<br /> | [[Q-derivative power rule]]<br /> | ||
| − | |||
=See Also= | =See Also= | ||
[[q-number|$q$-number]]<br /> | [[q-number|$q$-number]]<br /> | ||
| + | [[q-Pochhammer|$q$-Pochhammer]]<br /> | ||
=References= | =References= | ||
| − | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-number when a=n is a natural number|next=findme}}: (6.3) | + | * {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=Q-exponential E sub q|next=findme}} |
| + | * {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=q-number|next=findme}} | ||
| + | * {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(3.1)$ | ||
| + | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=Q-number when a=n is a natural number|next=findme}}: ($6.3$) | ||
{{:q-calculus footer}} | {{:q-calculus footer}} | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 04:23, 26 December 2016
The $q$-factorial is defined for a non-negative integer $k$ by $$[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
See Also
References
- D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next)
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions ... (previous) ... (next)
- 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(3.1)$
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.3$)
