EDIT: it seems implied by a response to this post that a proof of this fact does actually exist, but I've yet to see the proof or any intuition for the proof – so what is it?
OP:
According to a Wikipedia article on the Gamma Function, given a fixed integer $m$ and increasing integer $n$:
$$\lim_{n\to \infty} \frac{n!}{(n + m)!} \cdot (n + 1)^m = 1$$
I understand this part, however the article proceeds to say that (emphasis mine) "we do get a unique extension of the factorial function to the non-integers by insisting that this equation continue to hold when the arbitrary $m$ is replaced by an arbitrary complex number $z$."
$$\lim_{n\to \infty} \frac{n!}{(n + z)!} \cdot (n + 1)^z = 1, \forall z\in \mathbb{C} \space \ldots?$$
It seems to me that "insisting" is logically and mathematically weak. Why do mathematicians here get to insist this without explicitly proving it, especially considering how this result is used later in another proof? How does one know whether something is even fair to insist, without proof? It doesn't seem like an axiom or anything inherently assumable…
Thanks.
Best Answer
What the article means is there is a theorem, whose proof they haven't presented to you yet, that at least one generalization of the factorial function satisfies the desired condition. In fact, it's unique. There are also other theorems that highlight how this function is unique.