Hyperfactorial

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

Definition

The hyperfactorial of a positive integer n {\displaystyle n} {\displaystyle n} is the product of the numbers 1 1 , 2 2 , … , n n {\displaystyle 1^{1},2^{2},\dots ,n^{n}} {\displaystyle 1^{1},2^{2},\dots ,n^{n}}. That is,[1][2] 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).} {\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} {\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 K ( n + 1 ) = H ( n ) {\displaystyle K(n+1)=H(n)} {\displaystyle K(n+1)=H(n)}.[3]

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)\!,} {\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} {\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 p {\displaystyle 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}},} {\displaystyle H(p-1)\equiv (-1)^{(p-1)/2}(p-1)!!{\pmod {p}},} where ! ! {\displaystyle !!} {\displaystyle !!} is the notation for the double factorial.[4]

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

See also

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 11.22.33... nn", Messenger of Mathematics, 7: 43–47