Hyperfactorial

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]

Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with ${\displaystyle H(0)=1}$, is:^[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

for the factorials: where ${\displaystyle A\approx 1.28243}$ is the Glaisher–Kinkelin constant.^[2][5]

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

where ${\displaystyle !!}$ is the notation for the double factorial.^[4]

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

References

External links

• Weisstein, Eric W., "Hyperfactorial", MathWorld