## Three gamma function identities

Let $n=1,2,\ldots$ . Show that

$\sqrt{\pi}\;\Gamma (2n+1)=2^{2n}\Gamma\left( n+\dfrac{1}{2}\right) \Gamma (n+1)\qquad\left( 1\right)$

and

$\sqrt{\pi}\;\Gamma (2n)=2^{2n-1}\Gamma (n)\Gamma\left( n+\dfrac{1}{2}\right)\qquad \left( 2\right)$ .

Let $x\in\mathbb{R}$. If $x>0$, show that

$\sqrt{\pi}\;\Gamma (2x)=2^{2x-1}\Gamma (x)\Gamma\left( x+\dfrac{1}{2}\right)\qquad \left( 3\right)$.

Hints: for the first two identities use the formula proved here. As for the last one evaluate the beta function value B$(x,x)$ and by means of an appropriate  change of variable find a relation between B$(x,x)$ and B$\left(x,\dfrac{1}{2}\right)$.

PS. Listed in the Carnival of Mathematics #56. See pingback in the 1st comment.