In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:

• For $k, l \in \mathbb{Z}$, we have $k! \times l!$ divides $(k+l)!$.

• For any primitive polynomial $f(x) \in \mathbb{Z}[x]$ with $\deg f = k$ then $\mathrm{gcd}\{ f(a): a \in \mathbb{Z}\}$ divides $k!$

In the process of solving generalizing these two results, he invents a factorial for any set of integers $S \in \mathbb{Z}$. For any prime $p$, order the element of $S$ by:

• choose $a_0 \in S$
• find $a_1$ giving the smallest power of $(a_1 - a_0)$
• find $a_2$ giving the smallest power of $(a_2 - a_0)(a_2 - a_1)$
• ...
• find $a_k$ giving the smallest power of $\prod_{i< k} (a_k - a_i)$

One could look for analogues of the gamma function, stirling's approximation, binomial theorem and taylor series expansion of $e$ and indeed, Bhargava mentions these questions towards the end of the paper.

Have any of these questions been answered (partially or otherwise)?