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Gamma function and product
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Author:  Tolaso J Kos [ Thu Jul 14, 2016 10:14 am ]
Post subject:  Gamma function and product

Evaluate the product:

$$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$

Author:  Grigorios Kostakos [ Thu Jul 14, 2016 10:15 am ]
Post subject:  Re: Gamma function and product

Hello Tolis.

Using Gauss Multiplication Formula \[\displaystyle\prod_{k \mathop = 0}^{n - 1} \Gamma \Bigl({z + \frac k n}\Bigr) = ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n z} \Gamma({n z})\] for \(z=0\), we have that \begin{align*}
\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({\frac k n}\Bigr)&=\prod_{k \mathop = 1}^{n - 1} \Gamma \Bigl({0 + \frac k n}\Bigr) \\
&= ({2 \pi})^{\frac{n - 1}{2}} n^{\frac{1}{2} - n \cdot0} \Gamma({n \cdot0})\\
&= ({2 \pi})^{\frac{n - 1}{2}} \,\sqrt{n}\,.
\end{align*}

A proof of the Gauss Multiplication Formula can be found in here.

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