I was reading a book on the gamma function and it talked about an alternative proof of the duplication formula using the definition below:
$$G_{p}(x) = \frac{p!p^x}{(x)(x+1)...(x+p)}=\frac{p^x}{x(1+x/1)...(1+x/p)}$$
For $G(x)$ and $G(x+\frac{1}{2})$, then taking the product and then the limit as $p \to \infty$. I have tried to prove this with no success, any ideas?