I have the following expression
$$\binom{p-q}{\frac{p-q}{2}}-\binom{p-q}{\frac{p-q}{2}-1}=\frac{(p-q)!}{(\tfrac{p-q}{2})!(\tfrac{p-q}{2}+1)!},$$
where $p$ and $q$ are nonnegative integers with $p\geq q$. I have been trying to obtain an asymptotic formula for this expression as $p$ grows using Stirling's approximation:
$$\log{(n!)}= n\log{n}-n+\frac{1}{2}\log{(2\pi)}+\frac{1}{2}\log{n}+O(\tfrac{1}{n}).$$
Applying this approximation to this entire expression yields quite a large formula so I'll not reproduce it here, but every time I try and do this I end up with an asymptotic formula that is incorrect, usually by a factor of $e$. For example, consider the second factorial in the denominator:
$$\log{(((p-q)/2+1)!)}=((p-q)/2+1)\log{((p-q)/2+1)}-((p-q)/2+1)+\frac{1}{2}\log{(2\pi)}+\frac{1}{2}\log{((p-q)/2+1)}+O(\tfrac{1}{((p-q)/2+1)}).$$
Now as $p$ grows very large, what should this expression reduce to? Is it
$$(p/2)\log{(p/2)}-p/2+\frac{1}{2}\log{(2\pi)}+\frac{1}{2}\log{(p/2)}+O(\tfrac{1}{((p-q)/2)}),$$
or
$$((p-q)/2)\log{((p-q)/2)}-((p-q)/2)+\frac{1}{2}\log{(2\pi)}+\frac{1}{2}\log{((p-q)/2)}+O(\tfrac{1}{((p-q)/2)}),$$
or something else?