The factorial function and generalizations, extended

This paper presents an extension of Bhargava’s theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava’s factorials $k{!}_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge 2$, as well as “extreme” cases $b=1$ and $b=0$. It defines generalized factorials $k{!}_{S,T}$ and generalized binomial coefficients ${\left(\genfrac{}{}{0px}{}{k+\mathrm{\ell }}{k}\right)}_{S,T}$ as nonnegative integers, for all nonempty $S$ and allowing only $b$ in $T\subseteq \mathbb{N}$. It computes $b$-ordering invariants when $S$ is $\mathbb{Z}$ and when $S$ is the set of all primes.