Proof for Γ(p+1/2) using Double Factorial and nΓ(n)

[JKC]

Homework Statement

Prove that for a positive integer, p:

https://www.physicsforums.com/posts/5859454/I've tried this to little avail for the better part of an hour - I know there's a double factorial somewhere down the line but I've been unable to expand for the correct expression in terms of "p".

Homework Equations

Γ(p+1/2) = ((2p)!/4^p p!))√π

nΓ(n) = Γ(n+1)

The Attempt at a Solution

 

[fresh_42]

You are required to show us, what you've tried. Are you allowed to use Euler's reflection formula? The proofs I've found all uses it.

 

[JKC]

You are required to show us, what you've tried. Are you allowed to use Euler's reflection formula? The proofs I've found all uses it.

Yes the reflection formula is allowed. I tried applying it but wasn't able to solve. And sorry but writing out all these wrong workings would have taken quite some time. I will update the OP with some of my notes if there isn't anything when I wake up in a few hours.

 

[fresh_42]

You can use $\Gamma(p+\frac{1}{2}) = \Gamma ((p-\frac{1}{2}) + 1)$ and prove it with induction, because the functional equation gives you an expression with $\Gamma (p-\frac{1}{2})=\Gamma((p-1)+\frac{1}{2})$ for which the induction hypothesis applies. The reflection formula gives the induction base $(p=0)$, and the rest is some algebra with factorials.