# Definition:Factorial/Multiindices

## Definition

Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \geq 0$.

Then we define:

- $\displaystyle\alpha! = \prod_{j \mathop \in J} \alpha_j!$

where the factorial on the right is a factorial of natural numbers.

Note that by definition, all by finitely many of the $\alpha_j$ are zero, so the product over $J$ is convergent.

## Historical Note

The symbol $!$ used on $\mathsf{Pr} \infty \mathsf{fWiki}$ for the factorial, which is now universal, was introduced by Christian Kramp in his $1808$ work *Élémens d'arithmétique universelle*.

Before that, various symbols were used whose existence is now of less importance.

Notations for $n!$ in history include the following:

- $\sqbrk n$ as used by Euler
- $\mathop{\Pi} n$ as used by Gauss
- $\left\lvert {\kern-1pt \underline n} \right.$ and $\left. {\underline n \kern-1pt} \right\rvert$, once popular in England and Italy.

Augustus De Morgan declared his reservations about Kramp's notation thus:

*Amongst the worst barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation $n!$ ... which gives their pages the appearance of expressing admiration that $2$, $3$, $4$, etc., should be found in mathematical results.*

The use of $n!$ for non-integer $n$ is uncommon, as the Gamma function tends to be used instead.