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.
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.
The th superfactorial may be defined as: [1]
Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with , is: [1]
Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function. [2]
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. [3]
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 (the middle one, ) results in a square product. [4] 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. [1]
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