How to define fractional factorials, like 3.6!? [duplicate]

Solution 1:

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.)

Solution 2:

The factorial function $n! = \prod\limits_{i=1}^n i$ is defined only for positive integers $n$. But a standard move in function theory is to extend such functions to the reals, and then the complex plane. To do this, you have to decide what the logical "generalization" of the factorial function would be. An obvious place to start is the observation that the factorial obeys the recurrence relation $n! = n(n-1)!$, which has an obvious generalization in the reals, $f(x) = x\,f(x-1)$. (I'm calling this function $f$ now because it's no longer the factorial function.) We also need a fixed point; $1! = 1$, so set $f(1) = 1$. And it's convenient to shift it left by one: $f(x+1) = x\,f(x)$ (so now $f(x) = (x-1)!$ when $x$ is a positive integer.

There are still a great many functions that fit these constraints. But if you add just one more, for the function $f$ to be logarithmically convex over the positive reals, it's a theorem of Bohr and Mollerup (not that Bohr) that this uniquely defines the gamma function,

$$ \Gamma(t) = \int\limits_0^\infty x^{t-1} e^{-x} \;\text{d}x $$

So $\Gamma(t+1)$ is what mathematicians generally think of when you ask "what is the factorial of a real number that isn't an integer". However, what Excel does if asked to compute the factorial of a non-integer real is truncate it.