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 $n=1,2,3,\ldots$ or $n=\infty$, by | The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$ or $n=\infty$, by | ||
| − | $$(a;q)_n=\displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}).$$ | + | $$(a;q)_n=\left\{ |
| + | \begin{array}{ll} | ||
| + | 1, & \quad n=0 \\ | ||
| + | \displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}), & \quad n=1,2,3,\ldots \\ | ||
| + | \end{array} \right.$$ | ||
=Properties= | =Properties= | ||
=References= | =References= | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=findme|next=findme}}: $4.8 (1) (2)$ (assumes $|q|<1$; does not specifically say "$q$-shifted factorial") | ||
* {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-Gamma|next=Q-Gamma at z+1}} | * {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-Gamma|next=Q-Gamma at z+1}} | ||
* {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}} | * {{PaperReference|q-Special functions, a tutorial|1994|Tom H. Koornwinder|prev=findme|next=findme}} | ||
Revision as of 21:32, 17 June 2017
The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$ or $n=\infty$, by $$(a;q)_n=\left\{ \begin{array}{ll} 1, & \quad n=0 \\ \displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}), & \quad n=1,2,3,\ldots \\ \end{array} \right.$$
Properties
References
- 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.8 (1) (2)$ (assumes $|q|<1$; does not specifically say "$q$-shifted factorial")
- Daniel S. Moak: The q-gamma function for q greater than 1 (1980)... (previous)... (next)
- 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")
