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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, the hyperfactorial of a positive integer
is the product of the numbers of the form
from
to
.
Definition
The hyperfactorial of a positive integer
is the product of the numbers
. That is,
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 question ...
, 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. For ...
of hyperfactorials, beginning with
, is:
Interpolation and approximation
The hyperfactorials were studied beginning in the 19th century by
Hermann Kinkelin
Hermann Kinkelin (11 November 1832 – 1 January 1913) was a Swiss mathematician and politician.
Life
His family came from Lindau on Lake Constance. He studied at the Universities of Zurich, Lausanne, and Munich. In 1865 he became professor of ...
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) English ...
. As Kinkelin showed, just as the
factorials 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 ...
, the hyperfactorials can be continuously interpolated by the
K-function
In mathematics, the -function, typically denoted ''K''(''z''), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function.
Definition
Formally, the -function is define ...
.
Glaisher provided an
asymptotic formula
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as bec ...
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:
where
is the
Glaisher–Kinkelin constant In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted , is a mathematical constant, related to the -function and the Barnes -function. The constant appears in a number of sums and integrals, especially those ...
.
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 m ...
on the behavior of factorials modulo prime numbers, when
is an odd prime number
where the
is the notation for the
double factorial.
The hyperfactorials give the sequence of
discriminants of
Hermite polynomials 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