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
$$[k]_q!=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{k-1})=\dfrac{(q;q)_k}{(1-q)^k},$$
+
$$[n]_q! = [1]_q [2]_q \ldots [n]_q=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{n-1})=\dfrac{(q;q)_n}{(1-q)^n},$$
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.

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

$q$-calculus