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 | ||
| − | $$[a]_{n,q}= \displaystyle\prod_{k=0}^{n-1} [a+ | + | $$[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) ,$$ |
| + | where $[a]_q$ denotes a [[q-number|$q$-number]]. | ||
=Notes= | =Notes= | ||
Latest revision as of 21:07, 18 December 2016
The $q$-Pochhammer symbol $[a]_{n,q}$ is defined for $n=0$ by $[a]_{0,q}=1$, for $n=1,2,3,\ldots$ by the formula $$[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.
