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