In mathematics, and more specifically number theory, the hyperfactorial of a positive integer to

Definition

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

Interpolation and approximation

The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin and James Whitbread Lee Glaisher. 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.

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula is the Glaisher–Kinkelin constant.

Other properties

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.

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.

See also

• Superfactorial

References

External links