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 nth superfactorial may be defined as: 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. According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p is an odd prime number where !! is the notation for the double factorial. For every integer k, 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 (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, and equals the superfactorial when the given numbers are consecutive.