How to prove the half number factorial formula?

[laker88116]

Any ideas on how to prove this?
(n+\tfrac{1}{2})! = \sqrt{\pi} \prod_{k=0}^{n}\frac{2k+1}{2}

[shmoe]

You've presumably met the Gamma function. Have a look through identites involving Gamma that you know, you should find something useful.

[HallsofIvy]

How do you define (n+ 1/2)!???

(Schmo already told you that- my point is you should think about it!)

[laker88116]

Problem is, I don't know what Gamma is other than a greek letter. I can use the formula, that's not the problem. I just was curious if there was a way to prove it. I was messing with my calculator and I noticed that half numbers have factorials and other decimals don't. So, I looked this up. I am not sure what level math it is. I am through Calc 2. If you could let me know what these identies are, I would appreciate it.

[CRGreathouse]

Problem is, I don't know what Gamma is other than a greek letter. I can use the formula, that's not the problem. I just was curious if there was a way to prove it. I was messing with my calculator and I noticed that half numbers have factorials and other decimals don't. So, I looked this up. I am not sure what level math it is. I am through Calc 2. If you could let me know what these identies are, I would appreciate it.

\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}

For nonnegative integers, \Gamma(x+1)=x!. It is sensible to expand the notation x! to all real numbers other than the nonpositive integers.

[laker88116]

I'm not understanding that. What does it have to do with my equation I listed.

[shmoe]

Okie, you know that (and how) factorial n! is defined on the integers. We define Gamma as:

\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt

for all x>0 real numbers. Then you, having completed calc 2, can prove the following:

-the integral in the definition of Gamma does indeed converge when x>0
-for all x>0, \Gamma(x+1)=x\Gamma(x), the "Functional equation" of gamma
-if n is a non-negative integer, then n!=\Gamma(n+1), which justifies calling Gamma an extension of factorial.

Some calculators will use Gamma to make sense of arguments of x! that are not non-negative integers, and are essentially defining x!=\Gamma(x+1) when x is a real number greater than -1. I find it odd that yours does half integral values but not other decimals. I believe you, it just seems like an odd thing to do.

To prove your formula, you can use the identity involving sine CRGreathouse supplied to find \Gamma(1/2), then use the functional equation above to find Gamma at 3/2, 5/2, ... n/2 (using induction) then translate back to factorial to get the equation in your first post. Alternatively, look directly at the integral definition of \Gamma(1/2) and try to relate it back to the Gaussian probability integral.

[laker88116]

Alright, that makes sense, thanks.

[mathwonk]

Riemann observed in one of his early papers that this expression for factorials of non integers allows differentiation to non integral orders, since by the cauchy integral formula differentiation to a non integral order t, simply requires one to integrate a non integral power, no problem, and determine the appropriate non integral factorial to multiply the integral by.

so you can take the 1/2 derivative of something, e.g. or even the ith, I suppose.

this is a little industry today.

[rman144]

I have a somewhat related question. Is there/ does anyone know of a similar equation to the first one shown on this page for quarter integers. Basically:

(n+1/4)!

And particularly one that does not involve the gamma function.