Asymptotics of $\frac{n!}{(n- \alpha \log n)!}\left(\frac{c}{n}\right)^{\alpha \log n}$

Question

Let $c > 0$ be some constant, and consider for every positive integer $n$, the function: $$ f_\alpha(n) := \frac{n!}{(n- \lceil{\alpha \log n\rceil})!}\left(\frac{c}{n}\right)^{\alpha \log n}. $$ I would like to determine the smallest $\alpha^\star(c)$ such that $f_\alpha(n) = o(1)$ as $n \to \infty$, for $\alpha \geq \alpha^\star(c)$.

Using Stirling's approximation, \begin{align*} f_{\alpha}(n) &\sim \left(\frac{1}{1 - \alpha \frac{\log n}{n}}\right)^{n -\alpha \log n + \tfrac{1}{2}} n^{\alpha (\log c - 1) }. \end{align*} This right hand side seems a little difficult to bound. Any help would be appreciated!

Answer 1

Using the known asymptotics for the ratio of gamma functions (http://dlmf.nist.gov/5.11.E13), I obtained an asymptotics that is different from yours: $$ f_\alpha (n) = n^{\left\lceil {\alpha \log n} \right\rceil - \alpha \log n + \alpha \log c} \left( {1 + \mathcal{O}\!\left( {\frac{{\log ^2 n}}{n}} \right)} \right). $$ This shows that you need $$ \alpha < - \frac{1}{{\log c}} $$ to achieve $f_\alpha (n) \to 0$.