Hyperfactorial

Template:Short description Template:Use list-defined references In mathematics, and more specifically number theory, the hyperfactorial of a positive integer $n$ is the product of the numbers of the form $x^x$ from $1^1$ to $n^n$.

Definition

The hyperfactorial of a positive integer $n$ is the product of the numbers $1^1,2^2,\dots ,n^n$. That is,Template:R/superscriptTemplate:R/superscript $$H(n)=1^1\cdot 2^2\cdot \cdots n^n=\prod _{i=1}^n{i}^i=n^{n}H(n-1).$$ Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with $H(0)=1$, is:Template:R/superscript Template:Bi

Interpolation and approximation

The hyperfactorials were studied beginning in the 19th century by Hermann KinkelinTemplate:R/superscriptTemplate:R/superscript and James Whitbread Lee Glaisher.Template:R/superscriptTemplate:R/superscript 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 $K(n+1)=H(n)$.Template:R/superscript

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: $$H(n)=A{n}^{(6n^2+6n+1)/12}{e}^{-n^2/4}(1+\frac{1}{720n^2}-\frac{1433}{7257600n^4}+\cdots ),$$ where $A\approx 1.28243$ is the Glaisher–Kinkelin constant.Template:R/superscriptTemplate:R/superscript

Other properties

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $p$ is an odd prime number $$H(p-1)\equiv (-1{)}^{(p-1)/2}(p-1)!!(\mathrm{mod}p),$$ where $!!$ is the notation for the double factorial.Template:R/superscript

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.Template:R/superscript