multifatoriais

Multifatoriais, in Portuguese often referred to as multifactorials, are a family of functions that generalize the factorial by multiplying terms separated by a fixed step. For a fixed positive integer k and a nonnegative integer n, the multifatorial of order k is defined as

n!_(k) = n × (n − k) × (n − 2k) × ...,

where the product continues with positive terms until the last positive term remains. If k = 1, this reduces to the ordinary factorial n!. When k = 2, the object becomes the double factorial n!!, for example 7!! = 7 × 5 × 3 × 1 = 105 and 8!! = 8 × 6 × 4 × 2 = 384. For higher k, one obtains n!!!, n!!!!, and so on.

Notation and variants are common: n!_(k), n!^k, or n!^(k) are found in literature, with some sources using different stylistic forms. Multifactorials appear naturally in problems that involve products of integers in a fixed arithmetic progression, and they can play a role in certain recurrences and identities related to binomial-type coefficients.

Extensions to non-integer arguments can be formulated via gamma-function constructions, yielding generalized factorials of multifactorial type. These extensions are more advanced and depend on decomposing n modulo k, and they are discussed in references on generalized factorial functions and analytic combinatorics.

Historically, multifactorials have been studied as a straightforward generalization of the factorial concept, and they find use in number theory, combinatorics, and the study of special functions.