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

x

from

1

to

n

{\displaystyle n^{n}}

The hyperfactorial of a positive integer

is the product of the numbers

2

Following the usual convention for the empty product, the hyperfactorial of 0 is 1.

The sequence of hyperfactorials, beginning with

, is:[1] 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:

is the Glaisher–Kinkelin constant.

[2][5] According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when

is an odd prime number

( mod

is the notation for the double factorial.

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