multifactoriales

Multifactorials, also called k-factorials, are a generalization of the factorial operation. For a positive integer n and a fixed positive integer k, the k-factorial n!^(k) is the product of numbers from n down to the smallest positive term in steps of k: n × (n−k) × (n−2k) × ... . When k = 1 this reduces to the ordinary factorial.

Examples: 7!^(2) (often written 7!!) equals 7 × 5 × 3 × 1 = 105. 8!^(3) (written 8!!!) equals 8 × 5 × 2 = 80.

Notation and base cases: In many texts n!^(k) is denoted n!_(k) or n!^{(k)}. For 1 ≤ n ≤ k, n!^(k) = n. The general recurrence is n!^(k) = n × (n−k)!^(k) for n > k.

Relation to the gamma function: If n = qk + r with 0 ≤ r ≤ k−1, then for r > 0, n!^(k) = k^{q+1} Γ(q + 1 + r/k) / Γ(1 + r/k). If r = 0, n!^(k) = k^{q} q!.

Applications and connections: Multifactorials describe products over arithmetic progressions and arise in combinatorial identities and asymptotic analysis. They generalize the ordinary factorial (k = 1) and the double factorial (k = 2). They also relate to the gamma function, providing a way to extend factorial-like products to non-integer arguments for fixed step k.