In mathematics, and more specifically number theory, the superfactorial of a positive integer factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition

may be defined as: is the hyperfactorial.

Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence, is:

Properties

Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function for all nonnegative integers.

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime is an odd is the notation for the double factorial.

is a square number) results in a square product., and equals the superfactorial when the given numbers are consecutive.

References

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