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]
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:
According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when is an odd prime number
The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation. [1]
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]
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:
According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when is an odd prime number
The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation. [1]