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\otin\\{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.