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
Canonical name	GeneralizedFactorial
Date of creation	2013-03-22 16:08:31
Last modified on	2013-03-22 16:08:31
Owner	gilbert_51126 (14238)
Last modified by	gilbert_51126 (14238)
Numerical id	7
Author	gilbert_51126 (14238)
Entry type	Definition
Classification	msc 05A10