Difference between revisions of "Q-shifted factorial"

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The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$, by
 
The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$, by
$$\displaystyle\prod_{k=0}^{n-1} 1-aq^{k-1}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}).$$
+
$$(a;q)_n=\displaystyle\prod_{k=0}^{n-1} 1-aq^{k-1}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}).$$
  
 
=Properties=
 
=Properties=

Revision as of 02:56, 21 December 2016

The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$, by $$(a;q)_n=\displaystyle\prod_{k=0}^{n-1} 1-aq^{k-1}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}).$$

Properties

References