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 <math>n</math> is the product of the first <math>n</math> factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

The <math>n</math>th superfactorial <math>\mathit{sf}(n)</math> may be defined as:Template:R <math display=block>\begin{align} \mathit{sf}(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_{i=1}^{n} i! = n!\cdot\mathit{sf}(n-1)\\ &= 1^n \cdot 2^{n-1} \cdot \cdots n = \prod_{i=1}^{n} i^{n+1-i}.\\ \end{align}</math> Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with <math>\mathit{sf}(0)=1</math>, 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 <math>p</math> is an odd prime number <math display=block>\mathit{sf}(p-1)\equiv(p-1)!!\pmod{p},</math> where <math>!!</math> is the notation for the double factorial.Template:R

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

External links

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|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Superfactorial%7CSuperfactorial.html}} |title = Superfactorial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}