Inmathematics, 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 discriminantsofHermite polynomials in their probabilistic formulation.[1]