TITLE: How to define fractional factorials, like 3.6!?
QUESTION [12 upvotes]: I did not know that you could find an answer for that. However, I can only use Excel so far to do it. 
How to calculate 3.6! by hand?

REPLY [13 votes]: One way to do this is to notice that:
$$(n+t)!\approx n!n^t$$
when $n$ is large. For example, we have $(n+3)!=n!(n+1)(n+2)(n+3)\approx n!n^3$.
Also notice that:
$$t!=\frac{(t+n)!}{(t+1)(t+2)\dotsb(t+n)}$$
Putting these together, we have:
$$t!=\lim_{n\to\infty}\frac{n!n^t}{(t+1)(t+2)\dotsb(t+n)}$$
which is valid even when $t$ is not an integer!
In case you haven't seen it before: $\lim_{n\to\infty}$ means, roughly, that we let $n$ become larger and larger, and we see what value the expression gets closer and closer to.  For example, $\lim_{n\to\infty}\frac1n=0$, because as $n$ gets larger and larger (i.e. as $n$ "goes to infinity"), $\frac1n$ becomes closer and closer to $0$.
EDIT: What Excel does, however, is to simply truncate it. So, instead of being bothered to find out what 3.6! is (roughly 13.38129), it just gives you 3! instead (which is 6).
EDIT EDIT: I just realized that using this, we can prove the reflection formula $(-z)!z!=\pi z\csc(\pi z)$. (We need Euler's infinite product for the sine, though.)