Difference between revisions of "Q-factorial"
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The $q$-Factorial is defined for a non-negative integer $k$ by | The $q$-Factorial is defined for a non-negative integer $k$ by | ||
| − | $$[ | + | $$[n]_q! = [1]_q [2]_q \ldots [n]_q=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{n-1})=\dfrac{(q;q)_n}{(1-q)^n},$$ |
| − | where $(q;q)_k$ is the [[q-Pochhammer symbol]]. | + | where $[k]_q$ denotes a [[q-number|$q$-number]] and $(q;q)_k$ is the [[q-Pochhammer symbol]]. |
| + | |||
| + | =Properties= | ||
| + | {{:Q-derivative power rule}} | ||
| + | {{:Relationship between q-factorial and q-pochhammer}} | ||
{{:q-calculus footer}} | {{:q-calculus footer}} | ||
Revision as of 08:11, 3 May 2015
The $q$-Factorial is defined for a non-negative integer $k$ by $$[n]_q! = [1]_q [2]_q \ldots [n]_q=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{n-1})=\dfrac{(q;q)_n}{(1-q)^n},$$ where $[k]_q$ denotes a $q$-number and $(q;q)_k$ is the q-Pochhammer symbol.
Contents
Properties
Theorem
The following formula holds: $$D_q(z^n)=[n]_q z^{n-1},$$ where $D_q$ denotes the $q$-derivative and $[n]_q$ denotes the $q$-number.
Proof
References
- D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.2)$Relationship between q-factorial and q-pochhammer
