How to show that $ \prod\limits_{r=1}^{n}\Gamma{ \left({\frac {r}{n+1}}\right)}={\sqrt {\frac {(2\pi )^{n}}{n+1}}}$?

Question

I saw the following equation on Wikipedia, but I am not sure how to approach it.

$$ \prod_{r=1}^{n}\Gamma{ \left({\frac {r}{n+1}}\right)}={\sqrt {\frac {(2\pi )^{n}}{n+1}}}$$

Here are some other values listed on Wikipedia

$$ \prod _{r=1}^{2}\Gamma \left({\tfrac {r}{3}}\right)={\frac {2\pi }{\sqrt {3}}}$$ $$ \prod _{r=1}^{3}\Gamma \left({\tfrac {r}{4}}\right)={\sqrt {2\pi ^{3}}}$$ $$\prod _{r=1}^{4}\Gamma \left({\tfrac {r}{5}}\right)={\frac {4\pi ^{2}}{\sqrt {5}}}$$ Should I use induction for this problem? Any hints on where I should start on?

Answer 1

Consider Gauss Multiplication Formula in its general form

$$\forall z \notin \left\{{-\frac m n: m \in \mathbb N}\right\}: \prod_{k = 0}^{n - 1} \Gamma \left(z + \frac k n\right) = (2 \pi)^{(n - 1) / 2 }n^{1/2 - n z} \Gamma(n z)\tag1$$

Now by plugging in $z=1$ and $n=n+1$ we further get

$$\begin{align} \prod_{k = 0}^{(n+1)-1} \Gamma \left(1 + \frac k{(n+1)}\right) &= (2 \pi)^{((n+1) - 1) / 2} (n+1)^{1/2 - (n+1)\cdot 1} \Gamma((n+1)\cdot1)\\ \prod_{k = 1}^{n} \frac k{n+1}\Gamma \left(\frac k{n+1}\right) &= \sqrt{\frac{(2 \pi)^{n}}{n+1}} \frac{\Gamma(n+1)}{(n+1)^{n}}\\ \prod_{k = 1}^{n} \frac k{n+1}\prod_{k = 1}^{n}\Gamma \left(\frac k{n+1}\right) &= \sqrt{\frac{(2 \pi)^{n}}{n+1}} \frac{\Gamma(n+1)}{(n+1)^{n}}\\ \frac{n!}{(n+1)^n}\prod_{k = 1}^{n}\Gamma \left(\frac k{n+1}\right) &= \sqrt{\frac{(2 \pi)^{n}}{n+1}} \frac{\Gamma(n+1)}{(n+1)^{n}} \end{align}$$

$$\therefore~\prod_{k = 1}^{n}\Gamma \left(\frac k{n+1}\right) = \sqrt{\frac{(2 \pi)^{n}}{n+1}}$$

We started by utilizing the fundamental functional property of the Gamma Function $($note that the index $k$ has been uplifted to starting by $k=1$ due the fact that within the original product the first term $\Gamma(1)$ equals $1$ afterall and is so of minor matter$)$, followed by splitting up the product, rewriting the first subproduct in finite terms, and finally making use of the fact that $n!=\Gamma(n+1)$ for all $n\in\mathbb N$ .

Answer 2

Given the reflection formula for the $\Gamma$ function, $\Gamma(s)\Gamma(1-s)=\frac{\pi}{\sin(\pi s)}$, the problem immediately boils down to showing that $$ \prod_{k=1}^{n}\sin\left(\frac{\pi k}{n+1}\right) = \frac{2(n+1)}{2^{n+1}}, $$ which is well-known. One may use roots of unity or Chebyshev polynomials of the second kind, together with Vieta's formulas.