Gamma function and product
- Tolaso J Kos
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Gamma function and product
Evaluate the product:
$$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$
$$\Gamma \left ( \frac{1}{n} \right )\Gamma \left ( \frac{2}{n} \right )\cdots \Gamma \left ( \frac{n-1}{n} \right )$$
Imagination is much more important than knowledge.
- Grigorios Kostakos
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- Location: Ioannina, Greece
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.
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.
Grigorios Kostakos
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