# generalized factorial

Definition. The $[x,d]_{n}$ is defined for any number $x\in\mathbb{C}$, any stepsize $d\in\mathbb{C}$ and any $n\in\mathbb{Z}$, except for $-x\in\{d,2d,\dots,nd\}$, by

 $\displaystyle\displaystyle[x,d]_{n}:=\begin{cases}\prod_{j=0}^{n-1}(x-jd)&n\in% \mathbb{N}\\ 1&n=0\\ \prod_{j=1}^{-n}\frac{1}{x+jd}&-n\in\mathbb{N},-x\notin\{d,2d,\dots,nd\}.\end{cases}$

If $x=n$, $d=1$ and $n\in\mathbb{N}$ then

 $\displaystyle[n,1]_{n}=\begin{cases}n(n-1)\dots(2)(1)&n>0\\ 1&n=0,\end{cases}$

on which it follows that $[n,1]_{n}=n!$. This is why the above definition generalizes the notion of the usual factorial.

Title generalized factorial GeneralizedFactorial 2013-03-22 16:08:31 2013-03-22 16:08:31 gilbert_51126 (14238) gilbert_51126 (14238) 7 gilbert_51126 (14238) Definition msc 05A10