Difference between revisions of "Q-Pochhammer"
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| − | The $q$-Pochhammer symbol $ | + | 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) ,$$ |
| − | + | where $[a]_q$ denotes a [[q-number|$q$-number]]. | |
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| + | =Notes= | ||
| + | [http://mathworld.wolfram.com/q-PochhammerSymbol.html Mathworld] and [http://reference.wolfram.com/language/ref/QPochhammer.html Mathematica] define the "$q$-Pochhammer symbol" to be what we call the [[q-factorial|$q$-factorial]]. <br /> | ||
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{{:q-calculus footer}} | {{:q-calculus footer}} | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 21:07, 18 December 2016
The $q$-Pochhammer symbol $[a]_{n,q}$ is defined for $n=0$ by $[a]_{0,q}=1$, for $n=1,2,3,\ldots$ by the formula $$[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.
