A simple method for evaluating a product is term cancellation. For example, the product
$$\begin{align*}
\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\
&=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdots\frac{n-1}{n} = \frac{1}{n}
\end{align*}$$
is done via a telescoping argument. However, if we take a product that is just a bit more complicated, say
$$\prod_{k=1}^{n}\left(1 – \frac{1}{ak}\right)$$
for some $0,1 \neq a \in \mathbb{R}$, the argument immediately breaks down. I am interested in whether there exists a closed form solution for the general product given above.
This question is in part inspired by my attempt to prove the asymptotic bound here.
Best Answer
$$ \prod_{k=1}^{n}\left(1 - \frac{1}{ak}\right)=\frac{\Gamma(n+1-\frac1a)}{\Gamma(1-\frac1a)\Gamma(n+1)}=\frac1{n\mathrm B(n,1-\frac1a)} $$