Superfactorial Explained

is the product of the first

factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The

th superfactorial

sf(n)

may be defined as:

\begin\mathit(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_^ i! = n!\cdot\mathit(n-1)\\&= 1^n \cdot 2^ \cdot \cdots n = \prod_^ i^.\\\end

Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with

sf(0)=1

, is:

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

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when

is an odd prime number

\mathit(p-1)\equiv(p-1)!!\pmod,

where

is the notation for the double factorial.

For every integer

, 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. 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