Asymptotics on a class of Legendre formulas

Let $f$ be a real-valued function of a single variable such that it is positive over the primes. In this article, we construct a factorial, $n!_f$, associated to $f$, called the associated Legendre formula, or $f$-factorial, and show, subject to certain criteria, that $n!_f$ satisfies a weak Stirling approximation. As an application, we will give weak approximations to the Bhargava factorial over the set of primes and to a less well-known Legendre formula.