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
- Tom H. Koornwinder: q-Special functions, a tutorial (1994)... (previous)... (next)
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions ... (previous) ... (next) $(10.2.1)$ (does not specifically say "$q$-shifted factorial")
