Superfactorial
In mathematics, and more specifically number theory, the superfactorial of a positive integer 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 may be defined as:where is the hyperfactorial.
Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with, is:
Properties
Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as for all nonnegative integers.According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when is an odd prime number
where is the notation for the double factorial.
For every integer, the number is a square number. This may be expressed as stating that, in the formula for as a product of factorials, omitting one of the factorials results in a square product. Additionally, if any integers are given, the product of their pairwise differences is always a multiple of, and equals the superfactorial when the given numbers are consecutive.