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When I do some numerical test, I can get:

$$\frac{2^{2a+1}\Gamma(a+1)\Gamma(a+\frac32)}{\Gamma(2a+2)}=\sqrt{\pi}.$$

Here $a$ is to be taken such that the Gamma function well-defined.

My question is that is there is a way to prove this identity.

I tried to apply the identity $\Gamma(z+1)=z\Gamma(z)$ but no further result.

Thanks for any suggestions.

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This is known as the duplication formula for the Gamma function. It is commonly written as $$ \Gamma(2z) = \pi^{-1/2}2^{2z-1}\Gamma(z) \Gamma(z+1/2) $$ and appears as formula 5.5.5 in the NIST handbook.

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    $\begingroup$ Proven here & here. $\endgroup$
    – J.G.
    Mar 8 at 18:27

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