Difference between revisions of "Q-shifted factorial"

From specialfunctionswiki
Jump to: navigation, search
(References)
Line 9: Line 9:
  
 
=References=
 
=References=
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=findme|next=findme}}: $4.8 (1) (2)$ (assumes $|q|<1$)  
+
* {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=findme|next=Basic hypergeometric series phi}}: $4.8 (1) (2)$ (assumes $|q|<1$)  
 
* {{PaperReference|The q-gamma function for q greater than 1|1980|Daniel S. Moak|prev=Q-Gamma|next=Q-Gamma at z+1}}
 
* {{PaperReference|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 21:33, 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