I am studying a bit of asymptotics and for practice I decided to find the asymptotic of the following, $$s(n)=\sum_{i=0}^n\sqrt{i(n-i)}$$ as $n\to\infty$. This comes directly from this post, where @Gary provided some hints in the comments: $${\rm Li}_{-1/2}^2(z)=\left(\sum_{n=0}^\infty \sqrt nz^n\right)\left(\sum_{n=0}^\infty \sqrt n z^n\right)=\sum_{n=0}^\infty\underbrace{\sum_{i=0}^n \sqrt{i(n-i)}}_{s(n)}z^n$$ but I am unable to finish the problem with this method as I don't know how to find the $n$th Maclaurin coefficient. Feel free to also use any other methods to find the asymptotics (maybe Euler-Maclaurin), but such an answer will preferably come after the original issue is resolved. As a bonus, a derivation of the complete asymptotic expansion of $s(n)$ will also be nice to see.
Sorry for the long question. To outline:
- How to find the $n$th Maclaurin Coefficient?
- Alternative methods to find asymptotics?
- Complete asymptotic expansion?
My attempt is below.
By $(25.12.12)$, $$\begin{align*}{\rm{Li}}_{-1/2}(z)&=\frac{\sqrt\pi}{2}(-\log\left(z\right))^{-3/2}+\sum_{n=0}^\infty\frac{\zeta(-1/2-n)}{n!}\log^n(z) \end{align*}$$ which by Taylor's formula, $$\begin{align*} {\rm Li}_{-1/2}^2(z)&=-\frac{\pi}{4\log^3(z)}+\sum_{n=0}^\infty(-1)^n\sqrt\pi\frac{\zeta(-1/2-n)}{n!}(-\log(z))^{n-3/2}+F(z) \\ &=\frac{\pi}{4(1-z)}+\sum_{n=0}^\infty(-1)^n\sqrt\pi\frac{\zeta(-1/2-n)}{n!}(-\log(z))^{n-3/2}+G(z) \end{align*}$$ as $z\to1$, where $F(z),G(z)$ are holomorphic at $z=1$. Expanding the first term by its geometric series, $$\sum_{n=0}^\infty\left(s(n)-\frac{\pi}{4}\right)z^n=\sum_{n=0}^\infty(-1)^n\sqrt\pi\frac{\zeta(-1/2-n)}{n!}(-\log(z))^{n-3/2}+G(z)$$ as $z\to 1$. Now the RHS is, $$\sqrt\pi\zeta(-1/2)(-\log(z))^{-3/2}-\sqrt\pi\zeta(-3/2)(-\log(z))^{-1/2}+O\left(\sqrt{\log(z)}\right)$$ as $z\to1$, which by Taylor's formula, $$\sqrt\pi\zeta(-1/2)(1-z)^{-3/2}+\sqrt\pi\left(\zeta(-3/2)-\frac{3}{4}\zeta(-1/2)\right)(1-z)^{-1/2}+O\left((1-z)^{1/2}\right)$$ as $z\to1$. Now all that is left is to extract the $n$th Maclaurin coefficient of the function above as, $$s(n)=[z^n]{\rm Li}_{-1/2}^2(z).$$