Difference between revisions of "Q-Pochhammer"

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(Properties)
 
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$$(a;q)_n=\dfrac{(a;q)_{\infty}}{(aq^n;q)_{\infty}}\stackrel{n \in \mathbb{Z}^+}{=} \displaystyle\prod_{j=0}^{n-1} (1-aq^j)$$
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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
$$(a;q)_{\infty} = \displaystyle\prod_{j=0}^{\infty} (1-aq^j)$$
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$$[a]_{n,q}= \displaystyle\prod_{k=0}^{n-1} [a+k]_q  = \left(\dfrac{1-q^a}{1-q} \right) \left( \dfrac{1-q^{a+1}}{1-q} \right) \ldots \left( \dfrac{1-q^{a+n-1}}{1-q} \right) ,$$
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where $[a]_q$ denotes a [[q-number|$q$-number]].
  
$$(a;q)_{-n}=\dfrac{1} {(aq^{-n};q)_n} =\dfrac{1} {(1-aq^{-n})\ldots(1-aq^{-1})} = \dfrac{q^{\frac{n(n+1)}{2}}(-1)^n}{a^n (\frac{q}{a};q)_n}$$
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=Notes=
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[http://mathworld.wolfram.com/q-PochhammerSymbol.html Mathworld] and [http://reference.wolfram.com/language/ref/QPochhammer.html Mathematica] define the "$q$-Pochhammer symbol" to be what we call the [[q-factorial|$q$-factorial]]. <br />
  
=Properties=
 
{{:Relationship between q-factorial and q-pochhammer}}
 
{{:Relationship between Euler phi and q-Pochhammer}}
 
  
 
{{:q-calculus footer}}
 
{{:q-calculus footer}}
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[[Category:SpecialFunction]]

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