Asymptotics on a Class of Legendre Formulas

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