In mathematics, and more specifically

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, the hyperfactorial of a positive integer

n

is the product of the numbers of the form

x^x

from

1^1

n^n

Definition

The hyperfactorial of a positive integer

n

is the product of the numbers

1^1, 2^2, \dots n^n

. That is,

H(n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_^ i^i = n^n H(n-1).

Following the usual convention for the

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in questio ...

, the hyperfactorial of 0 is 1. The

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. Fo ...

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 James Whitbread Lee Glaisher FRS FRSE FRAS (5 November 1848, Lewisham – 7 December 1928, Cambridge), son of James Glaisher and Cecilia Glaisher, was a prolific English mathematician and astronomer. His large collection of (mostly) Engli ...

. As Kinkelin showed, just as the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...

s can be continuously interpolated by the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

, the hyperfactorials can be continuously interpolated by the K-function. Glaisher provided an asymptotic formula for the hyperfactorials, analogous to

Stirling's formula In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less ...

for the factorials:

H(n)=An^e^\left(1+\frac-\frac+\cdots\right),

where

A\approx 1.28243

is the Glaisher–Kinkelin constant.

Other properties

According to an analogue of

Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of ...

on the behavior of factorials modulo prime numbers, when

p

is an odd prime number

H(p-1)\equiv(-1)^(p-1)!!\pmod,

where the

!!

is the notation for the

double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...

. The hyperfactorials give the sequence of

discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the ori ...

s of

Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well ...

in their probabilistic formulation. * Interesting observation: the first 5 and 6 factors in the hyperfactorial generate the original names ADAVE and ADAVEL, respectively: www.ADAVE.name

References

External links

*{{MathWorld, id=Hyperfactorial, title=Hyperfactorial, mode=cs2 Integer sequences Factorial and binomial topics