I have a product of the form: $$ \frac{n}{n+a} \frac{mn-n}{m(n+a)-n} \frac{mn-2n}{m(n+a)-2n} \cdots \frac{mn-kn}{m(n+a)-kn}, $$ that is, $$ \prod_{i=0}^k \frac{mn-in}{m(n+a)-in}, $$ and I would like to know if there is a closed formula for it, or else an approximate formula. We may consider $m, n, a$ as constants, and it could be useful to know that $k << n$ and $k << m$.
Thanks a lot in advance.
$\displaystyle\prod\limits_{i=0}^k\frac{mn-in}{m(n+a)-in} = \frac{\prod\limits_{i=0}^k (m-i)}{ \prod\limits_{i=0}^k (m(1+a/n)-i) } = \frac{m!\,\Gamma(m(1+a/n)-k)}{(m-k-1)!\,\Gamma(m(1+a/n)+1)}$
$m!$ and $\Gamma(x)$ can be approximated by the Stirling formula.