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:
<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}\\
&=\frac{(n!)^{n+1{\prod_{i=1}^{n} i^{i = \frac{(n!)^{n+1{H(n)}
\end{align}</math>where <math>H</math> is the hyperfactorial.
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:
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 <math>sf(n) = G(n+2)</math> for all nonnegative integers.
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.
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. 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.
References
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