Difference between revisions of "Q-Pochhammer"

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The $q$-Pochhammer symbol $[a]_{n,q}$ is defined for $n=0$ by $[a]_{0,q}=1$, for $n=1,2,3,\ldots$ by the formula
 
The $q$-Pochhammer symbol $[a]_{n,q}$ is defined for $n=0$ by $[a]_{0,q}=1$, for $n=1,2,3,\ldots$ by the formula
$$[a]_{n,q}= \displaystyle\prod_{k=0}^{n-1} [a+m]_q.$$
+
$$[a]_{n,q}= \displaystyle\prod_{k=0}^{n-1} [a+k]_q = \left(\dfrac{1-q^a}{1-q} \right) \left( \dfrac{1-q^{a+1}}{1-q} \right) \ldots \left( \dfrac{1-q^{a+n-1}}{1-q} \right) ,$$
 
+
where $[a]_q$ denotes a [[q-number|$q$-number]].
=Properties=
 
[[q-Pochhammer as sum of q-binomial coefficients]]<br />
 
[[Relationship between q-factorial and q-pochhammer]]<br />
 
[[Relationship between Euler phi and q-Pochhammer]]<br />
 
  
 
=Notes=
 
=Notes=

Latest revision as of 21:07, 18 December 2016

The $q$-Pochhammer symbol $[a]_{n,q}$ is defined for $n=0$ by $[a]_{0,q}=1$, for $n=1,2,3,\ldots$ by the formula $$[a]_{n,q}= \displaystyle\prod_{k=0}^{n-1} [a+k]_q = \left(\dfrac{1-q^a}{1-q} \right) \left( \dfrac{1-q^{a+1}}{1-q} \right) \ldots \left( \dfrac{1-q^{a+n-1}}{1-q} \right) ,$$ where $[a]_q$ denotes a $q$-number.

Notes

Mathworld and Mathematica define the "$q$-Pochhammer symbol" to be what we call the $q$-factorial.


$q$-calculus