p-adic Valuations of Generalized Multifactorials: A Legendre-Type Formula

Abstract

The classical Legendre formula expresses the p-adic valuation of the factorial n! in terms of the base-p digit sum of n. We generalize this result to the α-th order multifactorial, the product of n and every α-th integer below it, for arbitrary integers α at least 2. When α divides n, the valuation reduces to a single closed form; in the general case, it is expressed as a Legendre-style sum over prime powers that counts divisibility conditions within the underlying arithmetic progression. Both cases are proved elementarily, using a structural identity relating the multifactorial to an ordinary factorial, together with basic properties of arithmetic progressions modulo prime powers.