In mathematics, and more specifically number theory, the superfactorial of a positive integer n {\displaystyle n} is the product of the first n {\displaystyle n} factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.
Definition The n {\displaystyle n} th superfactorial s f ( n ) {\displaystyle {\mathit {sf}}(n)} may be defined as:
s f ( n ) = 1 ! ⋅ 2 ! ⋅ ⋯ n ! = ∏ i = 1 n i ! = n ! ⋅ s f ( n − 1 ) = 1 n ⋅ 2 n − 1 ⋅ ⋯ n = ∏ i = 1 n i n + 1 − i = ( n ! ) n + 1 ∏ i = 1 n i i = ( n ! ) n + 1 H ( n ) {\displaystyle {\begin{aligned}{\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{aligned}}} where H {\displaystyle H} is the hyperfactorial. Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with s f ( 0 ) = 1 {\displaystyle {\mathit {sf}}(0)=1} , 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 s f ( n ) = G ( n + 2 ) {\displaystyle sf(n)=G(n+2)} for all nonnegative integers. According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number
s f ( p − 1 ) ≡ ( p − 1 ) ! ! ( mod p ) , {\displaystyle {\mathit {sf}}(p-1)\equiv (p-1)!!{\pmod {p}},}
… excerpt ends here. Continue reading the full article.
