Definition:Factorial/Multiindices
Definition
Let $\alpha$ be a multiindex, indexed by a set $J$ such that for each $j \in J$, $\alpha_j \ge 0$.
Then we define:
- $\ds \alpha! = \prod_{j \mathop \in J} \alpha_j!$
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.
In fact, Henry Ernest Dudeney was using $\left\lvert {\kern-1pt \underline n} \right.$ as recently as the $1920$s.
It can sometimes be seen rendered as $\lfloor n$.
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.