My question is how can one show that
$$\lim_{n \to \infty} \frac{1}{\binom{2n}{n}}\sum_{k=1}^n (-1)^k \binom{2n}{n+k}\frac{x^2}{x^2+\pi^2k^2} = \frac{1}{2}\Big(\frac{x}{\sinh(x)}-1\Big) $$
I find the proposed identity nice because it feels like we can look at the presence of the fraction in $x$ on the left hand side, for large $x$, as a perturbation to the textbook sum of alternating binomial coefficients with a little rearrangement.
I'm also generally interested in techniques to handle alternating sums of shifted central binomial sums with different weights, which prompted my question at MSE 2824529.
Skbmoore posted a proof of a neat and useful identity at 2827591 (and Tired commmented a slick proof) that $$ \,\,\,\,\,\sum_{k=1}^n (-1)^{k+1} \binom{2n}{n+k} k^s =
\binom{2n}{n} \sin(\pi s/2) \int_0^\infty \frac{dx \, \,x^s}{\sinh{\pi x}} \frac{n!^2}{(n+ix)!(n-ix)!}.
$$
Likely an apt modification of that formula will net us the asymptotics above, but I'm not quite clever enough with modifying the integrand to give me something nice in the limit of $s$ to zero.
Best Answer
Here is an approach:
We consider the following sum in the limit $n\rightarrow\infty$ (which we denote by $S$) :
We split the range of summation at some $\delta$ such that $1<<\delta<<n$ to get $S_n=S_{2,n}+S_{1,n}$ and observe that $k<<n$ in $[0,\delta]$ so we can expand the binomial around small $k/n$ which means that
$$ S_{1,n}=\sum_{k=0}^{\delta}(-1)^k\left(1+O(k^2/n^3)\right)\frac{1}{x^2+\pi^2 k^2} $$ since $|\sum_{k=0}^{\delta}(-1)^k\frac{k^2}{x^2+\pi^2 k^2}|<\sum_{k=0}^{\delta}\frac{k^2}{x^2+\pi^2 k^2}<\sum_{k=0}^{\delta}1=\delta$ we can bound the $O(k^2/n^3)$ term as follows
$$ S_{1,n}=\sum_{k=0}^{\delta}(-1)^k\frac{1}{x^2+\pi^2 k^2}+o(\delta/n^3) $$ Taking limits we get that $S_{1,n}$ approaches a constant to determined later
Next we show that the tail sum $S_{2,n}$ approaches zero so that the limit in question is indeed given by $(*)$. To this end, observe that on $[\delta,n]$ we have $\pi k >> x$ so we can expand the fraction around large $k$. We find
$$ S_{2,n}=\frac{1}{\binom{2n}{n}}\sum_{k=\delta}^n(-1)^k\binom{2n}{n+k}\left(\frac{1}{\pi^2 k^2}+O(1/k^4)\right) $$ since the binomial coefficient has a maximum at $k=0$ we can bound $\frac{\binom{2n}{n+k}}{\binom{2n}{n}}\leq1$ so
$$ |S_{2,n}|<\frac{1}{\pi^2}\sum_{k=\delta}^n\frac{1}{k^2}\sim \frac{1}{\pi^2} \int_{\delta}^n\frac{dk}{k^2}\sim \frac{1}{\pi^2\delta} $$ so, choosing $\delta$ to be a (slowly enough) increasing sequence of $n$ we have indeed that
Note that the $O(1/k^4)$ are vanishing even faster and can therefore also be neglected. Putting anything $(*)$ and $(**)$ together we get
were the last equality follows from Mittag-Lefflers Theorem (or the product expansion of sine)