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

Definition

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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

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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

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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]

References

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  1. ^ a b c Sloane, N. J. A. (ed.), "Sequence A002109 (Hyperfactorials: Product_{k = 1..n} k^k)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  2. ^ a b 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. ^ a b 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. ^ a b c 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. ^ a b Glaisher, J. W. L. (1877), "On the product 11.22.33... nn", Messenger of Mathematics, 7: 43–47
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