Superfactorial

Template:Short description Template:Use dmy dates Template:Use list-defined references In mathematics, and more specifically number theory, the superfactorial of a positive integer $n$ is the product of the first $n$ factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The $n$th superfactorial $𝑠𝑓(n)$ may be defined as:Template:R $$\begin{array}{cc}𝑠𝑓(n)& =1!\cdot 2!\cdot \cdots n!=\prod _{i=1}^{n}i!=n!\cdot 𝑠𝑓(n-1)\\ & =1^n\cdot 2^{n-1}\cdot \cdots n=\prod _{i=1}^n{i}^{n+1-i}.\\ \end{array}$$ Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with $𝑠𝑓(0)=1$, is:Template:R Template:Bi

Properties

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.Template:R

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number $$𝑠𝑓(p-1)\equiv (p-1)!!(\mathrm{mod}p),$$ where $!!$ is the notation for the double factorial.Template:R

For every integer $k$, the number $𝑠𝑓(4k)/(2k)!$ is a square number. This may be expressed as stating that, in the formula for $𝑠𝑓(4k)$ as a product of factorials, omitting one of the factorials (the middle one, $(2k)!$) results in a square product.Template:R Additionally, if any $n+1$ integers are given, the product of their pairwise differences is always a multiple of $𝑠𝑓(n)$, and equals the superfactorial when the given numbers are consecutive.Template:R

External links

• Template:MathWorld