There’s nothing special about the shift over by one in the Gamma function; there’s nothing special about it even in the integral that’s sometimes given for it. It somewhat annoys me that everyone keeps talking about it as though that arbitrary historical quirk shift by one is what’s key to extending the definition of factorial to fractions. (Bernoulli, Euler, etc., didn’t even start with that shift; it was introduced later for dumb reasons)
Anyway, for what it’s worth, here’s how the factorial function at values more general than whole numbers is defined:
Suppose you wanted to extend the factorial function to arbitrary arguments. How might you do it?
Well, of course, there are a million ways to do it. (Where “a million” = “infinitely many”). You could say the factorial function is the normal thing at natural numbers, and sqrt(7) everywhere else. This wouldn’t be a very useful extension, but it technically qualifies.
What would make a more useful extension, then? Well, we want an extension that preserves the key properties of the factorial function. For example, that n!/(n−1)! = n.
This still isn’t enough to pin down an extension, though. There’s still infinitely many extensions of that sort.
But there’s another interesting property of the factorial function: n!/(n−r)! is the number of ways to pick a sequence of r items from n choices, with no repetition. This is similar to, albeit less than, the number of ways to do it if you allow repetition, n[sup]r[/sup]. And as n gets larger and larger, the probability of repetition gets negligible; we find that the ratio between n!/(n−r)! and n[sup]r[/sup] approaches 1 as n grows large while r is held fixed. (For that matter, the same thing happens to the ratio between n−r and n).
In other words, if the difference between a and b is held fixed while their individual values grow large, then the ratio between a!/b! and b[sup]a−b[/sup] approaches 1.
This is a very useful property. If we continue to demand this for our extension, on top of the basic 0! = 1 and n!/(n−1)! = n, we will pin down a unique function, like so:
n! = n!/(n+d)! × (n+d)!/d! × d!
If d is a natural number, then n!/(n+d)! and d! are easy to calculate as rising products; combining these two factors produces 1/(n+1) × 2/(n+2) × … × d/(n+d).
Furthermore, our newest demand is that the middle factor, (n+d)!/d!, become replaceable with d[sup]n[/sup] as d grows large.
Thus, we have that n! is the limit, as the natural number d grows large, of d[sup]n[/sup] × 1/(n+1) × 2/(n+2) × … × d/(n+d). And this definition makes sense for all kinds of n, not just natural numbers. (When n is a negative integer, there will be a division by zero, but for all other complex numbers, this limit will be well-defined)
This defines the usual extension of the factorial function to arbitrary inputs. As we demonstrated, this is the unique way to do so while satisfying our key properties. As it turns out, other definitions accomplish the same effect (and therefore are equivalent to this one); for example, mathematicians will often note that one can define n! via “analytic continuation of the integral from x = 0 to infinity of x[sup]n[/sup]/e[sup]x[/sup] dx”. But there’s no need to introduce the general factorial via this complicated definition when the above simple definition is available instead.
One last note (reiterating what I said up top): mathematicians will also often talk about the so-called “Gamma (Γ) function”. The Gamma function is just this extension of the factorial function, shifted over by one. The shifting over by one is of no importance at all. It’s just a stupid historical convention. So don’t worry about it. All that actually matters is the argument above, constructing and establishing the uniqueness of a suitable interpretation of factorial for general (non-integer) inputs.
This works for fractions, complex numbers, even for matrices, all sorts of things. Its reciprocal converges to a finite value everywhere, even at negative integers (where it becomes zero); thus, we might also say the factorial is well-defined as reciprocal zero (infinity in the projectively extended numbers) at negative integers (and in similar ways for matrices or linear operators with negative integer eigenvalues…), and is well-defined as a finite quantity everywhere else.
