Definition:Factorial

Definition
Let $n \in \N$.

Then the factorial of $n$ is defined inductively as:
 * $n! = \begin{cases}

1 & : n = 0 \\ n \left({n - 1}\right)! & : n > 0 \end{cases}$

That is:
 * $n! = \displaystyle \prod_{k \mathop = 1}^n k = 1 \times 2 \times \cdots \times \left({n-1}\right) \times n$

where $\prod$ denotes product notation.

The first few factorials are:

$\begin{array}{r|r} n & n! \\ \hline 0 & 1 \\ 1 & 1 \\ 2 & 2 \\ 3 & 6 \\ 4 & 24 \\ 5 & 120 \\ 6 & 720 \\ 7 & 5 \ 040 \\ 8 & 40 \ 320 \\ 9 & 362 \ 880 \\ 10 & 3 \ 628 \ 800 \\ \end{array}$

...etc.

Definition for Multiindices
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 here, 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. For example, Gauss used $\Pi \left({n}\right)$ for $n!$.

Of major importance, however, is the Gamma function, which is an extension of the concept of the factorial to the complex plane.