Hyperfactorial
In mathematics, and more specifically number theory, the hyperfactorial of a positive integer [math]\displaystyle{ n }[/math] is the product of the numbers of the form [math]\displaystyle{ x^x }[/math] from [math]\displaystyle{ 1^1 }[/math] to [math]\displaystyle{ n^n }[/math].
Definition
The hyperfactorial of a positive integer [math]\displaystyle{ n }[/math] is the product of the numbers [math]\displaystyle{ 1^1, 2^2, \dots, n^n }[/math]. That is,[1][2] [math]\displaystyle{ 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]\displaystyle{ H(0)=1 }[/math], is:[1]
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.[3]
Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: [math]\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)\!, }[/math] where [math]\displaystyle{ A\approx 1.28243 }[/math] 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 [math]\displaystyle{ p }[/math] is an odd prime number [math]\displaystyle{ H(p-1)\equiv(-1)^{(p-1)/2}(p-1)!!\pmod{p}, }[/math] where [math]\displaystyle{ !! }[/math] is the notation for the double factorial.[4]
The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.[1]
References
- ↑ 1.0 1.1 1.2 Sloane, N. J. A., ed. "Sequence A002109 (Hyperfactorials: Product_{k = 1..n} k^k)". OEIS Foundation. https://oeis.org/A002109.
- ↑ 2.0 2.1 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
- ↑ 3.0 3.1 "Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechung" (in de), Journal für die reine und angewandte Mathematik 1860 (57): 122–138, 1860, doi:10.1515/crll.1860.57.122
- ↑ 4.0 4.1 4.2 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
- ↑ 5.0 5.1 "On the product 11.22.33... nn", Messenger of Mathematics 7: 43–47, 1877, https://archive.org/details/messengermathem01glaigoog/page/n56
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