LaTeX

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

Here’s the support page if you want to learn more.

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.

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