Hyperfactorial

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In mathematics, and more specifically number theory, the hyperfactorial of a positive integer is the product of the numbers of the form from to .

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Definition

The hyperfactorial of a positive integer is the product of the numbers . That is, [1] [2]

Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with , 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. [3]

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials:

where 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 is an odd prime number

where is the notation for the double factorial. [4]

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

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References

  1. 1 2 3 Sloane, N. J. A. (ed.), "SequenceA002109(Hyperfactorials: Product_{k = 1..n} k^k)", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation
  2. 1 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. 1 2 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. 1 2 3 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. 1 2 Glaisher, J. W. L. (1877), "On the product 11.22.33... nn", Messenger of Mathematics , 7: 43–47