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Hyperfactorial

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Number computed as a product of powers

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

Definition

The hyperfactorial of a positive integer n {\displaystyle n} is the product of the numbers 1 1 , 2 2 , … , n n {\displaystyle 1^{1},2^{2},\dots ,n^{n}} . That is, H ( n ) = 1 1 ⋅ 2 2 ⋅ ⋯ n n = ∏ i = 1 n i i = n n H ( n − 1 ) . {\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 H ( 0 ) = 1 {\displaystyle H(0)=1} , is:

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 and James Whitbread Lee Glaisher. 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.

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: H ( n ) = A n ( 6 n 2 + 6 n + 1 ) / 12 e − n 2 / 4 ( 1 + 1 720 n 2 − 1433 7257600 n 4 + ⋯ ) , {\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 A ≈ 1.28243 {\displaystyle A\approx 1.28243} is the Glaisher–Kinkelin constant.

Other properties

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

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.

References

  1. ^ Sloane, N. J. A. (ed.), "Sequence A002109 (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 1.2.3... n", Messenger of Mathematics, 7: 43–47

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