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 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 factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function as <math>K(n+1)= H(n)</math>.{{r|kinkelin}}

Glaisher provided an 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 modulo prime numbers, when <math>p</math> is an 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 discriminants of Hermite polynomials in their probabilistic formulation.{{r|oeis}}

==See also== *Superfactorial

==References== <references>

<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|1¹.2².3³... ''n''^''n''}} | 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 = 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>

</references>

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

Category:Integer sequences Category:Factorial and binomial topics