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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?

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1 Answer

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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.

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