Hyperfactorial

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

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 as ${\displaystyle K(n+1)=H(n)}$.^[3]

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: $${\displaystyle H(n)=An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}\left(1+{\frac {1}{720n^{2}}}-{\frac {1433}{7257600n^{4}}}+\cdots \right)\!,}$$ where ${\displaystyle A\approx 1.28243}$ is the Glaisher–Kinkelin constant.^[2][5]

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when ${\displaystyle p}$ is an odd prime number $${\displaystyle H(p-1)\equiv (-1)^{(p-1)/2}(p-1)!!{\pmod {p}},}$$ where ${\displaystyle !!}$ is the notation for the double factorial.^[4]

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.^[1]