Hyperfactorial Explained

n is the product of the numbers of the form x ^{x} from 1 ^{1} to

Definition

The hyperfactorial of a positive integer

n is the product of the numbers 1 ^{1,} 2 ^{2,} ... , n ^{n} . That is,$$H(n)\; =\; 1^1\backslash cdot\; 2^2\backslash cdot\; \backslash cdots\; n^n\; =\; \backslash prod\_^\; i^i\; =\; n^n\; H(n-1).$$Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with H( 0 ) =1 , is:

Interpolation and approximation

The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin and James Whitbread Lee Glaisher. 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.

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials:$$H(n)\; =\; An^e^\backslash left(1+\backslash frac-\backslash frac+\backslash cdots\backslash right)\backslash !,$$where

A ≈ 1 . 28243 is the Glaisher–Kinkelin constant.

Other properties

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when

p is an odd prime number$$H(p-1)\backslash equiv(-1)^(p-1)!!\backslash pmod,$$where !! is the notation for the double factorial.

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.