Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$
Solution 1:
$$ \prod_{k=1}^{n}\left(1 - \frac{1}{ak}\right)=\frac{\Gamma(n+1-\frac1a)}{\Gamma(1-\frac1a)\Gamma(n+1)}=\frac1{n\mathrm B(n,1-\frac1a)} $$
Solution 2:
$$\prod_{k=1}^n\left(1 - \frac1{ak}\right)=\frac{\prod\limits_{k=1}^n \left(k-\frac1{a}\right)}{n!}=\frac{\prod\limits_{k=0}^{n-1} \left(-\frac1{a}+k+1\right)}{n!}=\frac{\left(1-\frac1{a}\right)_n}{n!}$$
where $(a)_n=\prod\limits_{j=0}^{n-1}(a+j)=\frac{\Gamma(a+n)}{\Gamma(a)}$ is the Pochhammer symbol. (Essentially, the second expression in @Did's answer.)