Number computed as a product of powers
In
mathematics, and more specifically
number theory, 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,
[1]
[2]
![{\displaystyle H(n)=1^{1}\cdot 2^{2}\cdot \cdots n^{n}=\prod _{i=1}^{n}i^{i}=n^{n}H(n-1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5209a492b1f1e9fdaa9a792b61c47990fc4e03)
Following the usual convention for the
empty product, the hyperfactorial of 0 is 1. The
sequence of hyperfactorials, beginning with
![{\displaystyle H(0)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40545adb3fc1a115193c0391f29a86662d5925c)
, is:
[1]
1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (sequence
A002109 in the
OEIS)
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
asymptotic formula for the hyperfactorials, analogous to
Stirling's formula for the factorials:
![{\displaystyle H(n)=An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}\left(1+{\frac {1}{720n^{2}}}-{\frac {1433}{7257600n^{4}}}+\cdots \right)\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb6d02f0b998ec9fe942f808f6f5681fbb7164bd)
where
![{\displaystyle A\approx 1.28243}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22a3bc7b72ba967329155ce55cb9365f82a59e0c)
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
is an
odd prime number
![{\displaystyle H(p-1)\equiv (-1)^{(p-1)/2}(p-1)!!{\pmod {p}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb53ae8443557f30d38ecdde09f432c0add2b12f)
where
![{\displaystyle !!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6e2480ece878ba9a96d09f1fe710c7117f82f8)
is the notation for the
double factorial.
[4]
The hyperfactorials give the sequence of
discriminants of
Hermite polynomials in their probabilistic formulation.
[1]
References
- ^
a
b
c
Sloane, N. J. A. (ed.),
"Sequence A002109 (Hyperfactorials: Product_{k = 1..n} k^k)", The
On-Line Encyclopedia of Integer Sequences, OEIS Foundation
- ^
a
b Alabdulmohsin, Ibrahim M. (2018), Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Cham: Springer, pp. 5–6,
doi:
10.1007/978-3-319-74648-7,
ISBN
978-3-319-74647-0,
MR
3752675,
S2CID
119580816
- ^
a
b
Kinkelin, H. (1860), "Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung" [On a transcendental variation of the gamma function and its application to the integral calculus],
Journal für die reine und angewandte Mathematik (in German), 1860 (57): 122–138,
doi:
10.1515/crll.1860.57.122,
S2CID
120627417
- ^
a
b
c Aebi, Christian; Cairns, Grant (2015), "Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials",
The American Mathematical Monthly, 122 (5): 433–443,
doi:
10.4169/amer.math.monthly.122.5.433,
JSTOR
10.4169/amer.math.monthly.122.5.433,
MR
3352802,
S2CID
207521192
- ^
a
b
Glaisher, J. W. L. (1877),
"On the product 11.22.33... nn",
Messenger of Mathematics, 7: 43–47
External links