polyfactorial

Polyfactorial is an informal umbrella name for several families of functions that generalize the ordinary factorial. The term is not uniquely defined in the literature; common members of the class include multifactorials (k-factorials), superfactorials, hyperfactorials and iterated factorials, each arising from a different natural extension of the product n!.

A commonly used generalization is the multifactorial or k-factorial, which for a positive integer k is defined by the descending product n!_(k) = n (n − k) (n − 2k) … stopping at the last positive term. Double factorial (n!!) and triple factorial are special cases. Multifactorials satisfy simple recurrence relations n!_(k) = n · (n − k)!_(k) and reduce to the ordinary factorial when k = 1.

Other forms called polyfactorials include the superfactorial, typically defined as the product of factorials ∏_{j=1}^n j!, and the hyperfactorial H(n) = ∏_{j=1}^n j^j. These products are closely related to higher-order special functions: Barnes G-function encodes products of factorials, while the Gamma function and its generalizations represent factorial-like growth continuously over the complex plane.

Polyfactorial-type functions feature in combinatorics, special-function theory and asymptotic analysis. They have explicit product definitions, satisfy recurrence or functional equations, and admit asymptotic expansions analogous to Stirling’s formula. Applications include counting problems with layered structure, evaluations of determinants and moment formulas in probability. Because the term “polyfactorial” is used variably, precise meaning in any context is normally clarified by the defining product or recurrence.