View source for Hyperfactorial

{{Short description|Number computed as a product of powers}}
{{Use dmy dates|cs1-dates=ly|date=December 2021}}
{{Use list-defined references|date=December 2021}}
In [[mathematics]], and more specifically [[number theory]], the '''hyperfactorial'''  of a positive [[integer]] <math>n</math> is the product of the numbers of the form <math>x^x</math> from <math>1^1</math> to {{nowrap|<math>n^n</math>.}}

==Definition==
The '''hyperfactorial'''  of a positive integer <math>n</math> is the product of the numbers <math>1^1, 2^2, \dots, n^n</math>. That is,{{r|oeis|summability}}
<math display=block>
H(n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_{i=1}^{n} i^i = n^n H(n-1).</math>
Following the usual convention for the [[empty product]], the hyperfactorial of 0 is 1. The [[integer sequence|sequence]] of hyperfactorials, beginning with <math>H(0)=1</math>, is:{{r|oeis}}
{{bi|left=1.6|1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... {{OEIS|A002109}}}}

==Interpolation and approximation==
The hyperfactorials were studied beginning in the 19th century by [[Hermann Kinkelin]]{{r|kinkelin|wilson}} and [[James Whitbread Lee Glaisher]].{{r|glaisher|wilson}} As Kinkelin showed, just as the [[factorial]]s can be [[continuous function|continuously]] interpolated by the [[gamma function]], the hyperfactorials can be continuously interpolated by the [[K-function]].{{r|kinkelin}}

Glaisher provided an [[asymptotic analysis|asymptotic]] formula for the hyperfactorials, analogous to [[Stirling's formula]] for the factorials:
<math display=block>H(n) = An^{(6n^2+6n+1)/12}e^{-n^2/4}\left(1+\frac{1}{720n^2}-\frac{1433}{7257600n^4}+\cdots\right)\!,</math>
where <math>A\approx 1.28243</math> is the [[Glaisher–Kinkelin constant]].{{r|summability|glaisher}}

==Other properties==
According to an analogue of [[Wilson's theorem]] on the behavior of factorials [[modular arithmetic|modulo]] [[prime number|prime]] numbers, when <math>p</math> is an [[parity (mathematics)|odd]] prime number
<math display=block>H(p-1)\equiv(-1)^{(p-1)/2}(p-1)!!\pmod{p},</math>
where <math>!!</math> is the notation for the [[double factorial]].{{r|wilson}}

The hyperfactorials give the sequence of [[discriminant]]s of [[Hermite polynomials]] in their probabilistic formulation.{{r|oeis}}

==References==
{{reflist|refs=

<ref name=oeis>{{cite OEIS|1=A002109|2=Hyperfactorials: Product_{k = 1..n} k^k|mode=cs2}}</ref>

<ref name=glaisher>{{citation
 | last = Glaisher | first = J. W. L. | author-link = James Whitbread Lee Glaisher
 | journal = [[Messenger of Mathematics]]
 | pages = 43–47
 | title = On the product {{math|1<sup>1</sup>.2<sup>2</sup>.3<sup>3</sup>... ''n''<sup>''n''</sup>}}
 | url = https://archive.org/details/messengermathem01glaigoog/page/n56
 | volume = 7
 | year = 1877}}</ref>

<ref name=kinkelin>{{citation
 | last = Kinkelin | first = H. | author-link = Hermann Kinkelin
 | doi = 10.1515/crll.1860.57.122
 | journal = [[Crelle's Journal|Journal für die reine und angewandte Mathematik]]
 | language = de
 | pages = 122–138
 | title = Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung
 | trans-title = On a transcendental variation of the gamma function and its application to the integral calculus
 | volume = 1860
 | year = 1860| issue = 57 | s2cid = 120627417 }}</ref>

<ref name=summability>{{citation
 | last = Alabdulmohsin | first = Ibrahim M.
 | doi = 10.1007/978-3-319-74648-7
 | isbn = 978-3-319-74647-0
 | location = Cham
 | mr = 3752675
 | pages = 5–6
 | publisher = Springer
 | title = Summability Calculus: A Comprehensive Theory of Fractional Finite Sums
 | year = 2018| s2cid = 119580816
 }}</ref>

<ref name=wilson>{{citation
 | last1 = Aebi | first1 = Christian
 | last2 = Cairns | first2 = Grant
 | doi = 10.4169/amer.math.monthly.122.5.433
 | issue = 5
 | journal = [[The American Mathematical Monthly]]
 | jstor = 10.4169/amer.math.monthly.122.5.433
 | mr = 3352802
 | pages = 433–443
 | title = Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials
 | volume = 122
 | year = 2015| s2cid = 207521192
 }}</ref>

}}

==External links==
*{{MathWorld|id=Hyperfactorial|title=Hyperfactorial|mode=cs2}}

[[Category:Integer sequences]]
[[Category:Factorial and binomial topics]]