## LaTeX

I just learned that I can post LaTeX in WordPress (sorry, I’m slow).

Testing…

The sum $S(s,n) = \sum_{k=0}^{2n} (-1)^{k+n} \binom{2n}{k}^s$ can be expressed as the asymptotic series

$\displaystyle S(s,n) = 2 \cdot (-1)^n \sum_{k=0}^{h} (-1)^k \binom{2n}{k}^s + O(n^{(h+1)s})$

in the case where $s<0$ and $h$ are fixed and $n \rightarrow \infty$.

In the case $s>0$, we may apply the residue theorem to replace the sum by the integral

$\displaystyle S(s,n) = \int_C \left( \frac{\Pi(2n)}{\Pi(n+z) \Pi(n-z)} \right)^s \frac{dz}{2i \sin \pi z}$

for some suitably chosen integration path $C$.