Template:Short description Template:Use dmy dates Template:Use list-defined references 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 Template:Nowrap
DefinitionEdit
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,Template:R <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:Template:R Template:Bi
Interpolation and approximationEdit
The hyperfactorials were studied beginning in the 19th century by Hermann KinkelinTemplate:R and James Whitbread Lee Glaisher.Template:R 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.Template:R
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.Template:R
Other propertiesEdit
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.Template:R
The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.Template:R
ReferencesEdit
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Hyperfactorial%7CHyperfactorial.html}} |title = Hyperfactorial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}