TITLE: How to prove that $\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}$?
QUESTION [5 upvotes]: My task is to show that: 
$$\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}.$$
I'm trying to show this using properties of the gamma function, but it seams every step I take doesn't take me in the right direction. Any suggestions?

REPLY [3 votes]: From the functional equation
$$\frac12\sqrt{\pi}=\frac12\Gamma\left(\frac12\right)=\Gamma\left(\frac32\right)$$
Again from the functional equation,
$$\Gamma\left(\frac{11}6\right)=\frac56\Gamma\left(\frac56\right)$$
From the duplication formula,
$$\Gamma\left(\frac13\right)\Gamma\left(\frac56\right)=\sqrt[3]2\sqrt{\pi}\Gamma\left(\frac23\right)$$
And then from the reflection formula,
$$\Gamma\left(\frac13\right)\Gamma\left(\frac23\right)=\frac{2\pi}{\sqrt3}$$
Multiply all those equations together to get
$$\frac12\sqrt{\pi}\left(\Gamma\left(\frac13\right)\right)^2\Gamma\left(\frac{11}6\right)\Gamma\left(\frac56\right)\Gamma\left(\frac23\right)=\frac{10\pi^{\frac32}\sqrt[3]2}{6\sqrt3}\Gamma\left(\frac32\right)\Gamma\left(\frac56\right)\Gamma\left(\frac23\right)$$
Simplify, and that should do it.