Multifactorials

Multifactorials are a generalization of the factorial operation that extends the product to steps larger than one. For a positive integer n and a positive integer k, the k-fold or multifactorial of n is defined as n!(k) = n × (n − k) × (n − 2k) × ..., with the product continuing until the next factor would be nonpositive. Equivalently, n!(k) can be written as the product ∏_{i=0}^{m−1} (n − i k), where m = ⌈n/k⌉.

Notation and special cases

The multifactorial is sometimes written as n!(k) or n!^{(k)}. When k = 1, it reduces to the ordinary factorial n!. When k = 2, it becomes the double factorial n!!.

Examples

7!(3) = 7 × 4 × 1 = 28; 6!(3) = 6 × 3 = 18; 6!(4) = 6 × 2 = 12; 5!(4) = 5 × 1 = 5.

Properties and connections

If n is a multiple of k, say n = m k, then n!(k) = k^m m!. In general there is no simple closed form, but the multifactorial can be related to the gamma function and expressed in terms of products over arithmetic progressions. For certain values of n and k, these expressions facilitate analytic or asymptotic analyses.

See also

Factorial, Double factorial, Triple factorial.