In mathematics, and more specifically number theory, the hyperfactorial of a positive integer ${\displaystyle n}$ is the product of the numbers of the form ${\displaystyle x^{x))$ from ${\displaystyle 1^{1))$ to ${\displaystyle n^{n))$.

Definition

The hyperfactorial of a positive integer ${\displaystyle n}$ is the product of the numbers ${\displaystyle 1^{1},2^{2},\dots ,n^{n))$. That is,^[1][2]

$${\displaystyle H(n)=1^{1}\cdot 2^{2}\cdot \cdots n^{n}=\prod _{i=1}^{n}i^{i}=n^{n}H(n-1).}$$

${\displaystyle H(0)=1}$

Interpolation and approximation

The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin^[3][4] and James Whitbread Lee Glaisher.^[5][4] As Kinkelin showed, just as the factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function.^[3]

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials:

$${\displaystyle H(n)=An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}\left(1+{\frac {1}{720n^{2))}-{\frac {1433}{7257600n^{4))}+\cdots \right)\!,}$$

${\displaystyle A\approx 1.28243}$

Other properties

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when ${\displaystyle p}$ is an odd prime number

$${\displaystyle H(p-1)\equiv (-1)^{(p-1)/2}(p-1)!!{\pmod {p)),}$$

${\displaystyle !!}$

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.^[1]