Question 10.46: Prove that for m = 1, 2, 3, ... , Γ(1/m)Γ(2/m)Γ(3/m) ··· Γ(m...
Prove that for m=1,2,3, \ldots,
\Gamma\left(\frac{1}{m}\right) \Gamma\left(\frac{2}{m}\right) \Gamma\left(\frac{3}{m}\right) \cdots \Gamma\left(\frac{m-1}{m}\right)=\frac{(2 \pi)^{(m-1) / 2}}{\sqrt{m}}
We have
P=\Gamma\left(\frac{1}{m}\right) \Gamma\left(\frac{2}{m}\right) \cdots \Gamma\left(1-\frac{1}{m}\right)=\Gamma\left(1-\frac{1}{m}\right) \Gamma\left(1-\frac{2}{m}\right) \cdots \Gamma\left(\frac{1}{m}\right)
Then, multiplying these products term by term and using Problem 10.13, page 337, and Problem 1.52, page 32, we find
\begin{aligned} P^{2} & =\left\{\Gamma\left(\frac{1}{m}\right) \Gamma\left(1-\frac{1}{m}\right)\right\}\left\{\Gamma\left(\frac{2}{m}\right) \Gamma\left(1-\frac{2}{m}\right)\right\} \cdots\left\{\Gamma\left(1-\frac{1}{m}\right) \Gamma\left(\frac{1}{m}\right)\right\} \\ & =\frac{\pi}{\sin (\pi / m)} \cdot \frac{\pi}{\sin (2 \pi / m)} \cdots \frac{\pi}{\sin (m-1) \pi / m} \\ & =\frac{\pi^{m-1}}{\sin (\pi / m) \sin (2 \pi / m) \cdots \sin (m-1) \pi / m}=\frac{\pi^{m-1}}{m / 2^{m-1}}=\frac{(2 \pi)^{m-1}}{m} \end{aligned}
or P=(2 \pi)^{(m-1) / 2} / \sqrt{m}, as required.