When is it not okay to substitute $n! = \Gamma(n + 1)$?

Question

Let's say you just derived some formula that included integer parameters $m$ and $n$, and you wound up with something that had $m!$ or $n!$ or something similar in it, like the following example: $$ \operatorname{I}(m,\ n\ |\ x) = \int_0^x{t^m(1-t)^n\text{d}t} = m!n!\sum_{k=0}^{n}{(-1)^k\frac{x^{m+k+1}(1-x)^{n-k}}{(n-k)!(m+k+1)!}}, \quad m, n \in \Bbb{N} $$ Is it perfectly reasonable to say $$ \operatorname{I}(m,\ n\ |\ x) = \Gamma(m+1)\Gamma(n+1)\sum_{k=0}^{n}{(-1)^k\frac{x^{m+k+1}(1-x)^{n-k}}{\Gamma(n-k+1)\Gamma(m+k+2)}}, \quad m \in \Bbb{R}? $$ My intuition says yes because there is nothing in the equation which requires that either $m$ to be an integer ($n$ must be an integer for the sum to be finite as Yves Daoust pointed out). If it were something like $$ \text{something} = \sum_{k=0}^{n!}{\text{another thing}} $$ then obviously it would make no sense to say $$ \text{something} = \sum_{k=0}^{\Gamma(n+1)}{\text{another thing}} $$ So in general, if the only thing in an equation that requires a parameter to be an integer is factorial, then is it perfectly fine to replace it with the gamma function? Or are there very rare and tricky scenarios where using the gamma function would be false but using factorial would be true?

Answer 1

Once I saw a notation for $a\ge 0$ $$a! = \prod_{n\in \Bbb N,\,n< \lfloor a\rfloor}(a-n).$$ In this case, $a!\ne \Gamma(a+1)$ for non-integer $a$.

If we don't count this extreme case of redefinition of factorial, I don't really see a scenario where we can't replace factorial by gamma-function.

Answer 2

Here is a contrived example. The formula $n! = \cos(2\pi n) \Gamma(n+1)$ is true for all non-negative integers, but it isn't true for non-integer values of $n$.