Hyperfactorial

In mathematics, and more specifically number theory, the hyperfactorial of a positive integer ${\displaystyle n}$ is the product of the numbers of the form ${\displaystyle x^{x}}$ from ${\displaystyle 1^{1}}$ to ${\displaystyle n^{n}}$.

Definition

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]

1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (sequence A002109 in the OEIS)

Interpolation and approximation

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]}

Other properties

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]

References

1. Sloane, N. J. A. (ed.), "SequenceA002109(Hyperfactorials: Product_{k = 1..n} k^k)", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation
2. Alabdulmohsin, Ibrahim M. (2018), Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Cham: Springer, pp. 5–6, doi:10.1007/978-3-319-74648-7, ISBN   978-3-319-74647-0, MR   3752675, S2CID   119580816
3. Kinkelin, H. (1860), "Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung" [On a transcendental variation of the gamma function and its application to the integral calculus], Journal für die reine und angewandte Mathematik (in German), 1860 (57): 122–138, doi:10.1515/crll.1860.57.122, S2CID   120627417
4. Aebi, Christian; Cairns, Grant (2015), "Generalizations of Wilson's theorem for double-, hyper-, sub- and superfactorials", The American Mathematical Monthly , 122 (5): 433–443, doi:10.4169/amer.math.monthly.122.5.433, JSTOR   10.4169/amer.math.monthly.122.5.433, MR   3352802, S2CID   207521192
5. Glaisher, J. W. L. (1877), "On the product 1¹.2².3³... nⁿ", Messenger of Mathematics , 7: 43–47

External links

• Weisstein, Eric W., "Hyperfactorial", MathWorld