TITLE: What is the definition of $2.5!$? (2.5 factorial)
QUESTION [5 upvotes]: I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?
I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$.

REPLY [7 votes]: The other answers have already given nice explanations, so I shall instead be teaching a different lesson: as always, when confronted with seemingly peculiar behavior in your device, taking a look at your fine device's fine manual is almost always a profitable first step. As it turns out, looking at page 58 of your fine calculator's fine manual mentions the following:

So yes, do read the fine manual. ;)

As a further addition: there is what is called Gauss's duplication formula, which in factorial form goes as
$$\left(n+\frac12\right)!=\frac{\sqrt{\pi}(2n+2)!}{4^{n+1}(n+1)!}$$
This allows you to express factorials of semi-integers in terms of factorials and $\left(-\frac12\right)!=\sqrt \pi$.