In my research work, the following two series come out describing the 1st and 2nd order of a physical phenomenon:
$$\mathfrak{f}_1(\alpha )=2\cdot\sum_{k=1}^{+\infty} \ \begin{cases} \displaystyle\frac{1}{64k^2}, & |k|=\displaystyle\frac{1}{2\alpha}\\ \displaystyle\frac{\cos(\alpha\pi k)^2}{4(\pi k)^2((2\alpha k)^2-1)^2}, & |k|\neq\displaystyle\frac{1}{2\alpha} \end{cases} , $$
$$\mathfrak{f}_2(\alpha )=2\cdot\sum_{k=1}^{+\infty} \begin{cases} \displaystyle\frac{9}{256\pi^2k^4}, & |k|=\displaystyle\frac{1}{2\alpha} \\ \displaystyle\left( \frac{\alpha\sin(\alpha\pi k)}{2(\pi k)((2\alpha k)^2-1)}+ \displaystyle\frac{(3(2\alpha k)^2-1)\cos(\alpha\pi k)}{2(\pi k)^2((2\alpha k)^2-1)^2} \right)^2, & |k|\neq\displaystyle\frac{1}{2\alpha}\\ \end{cases} , $$
where $0\leq\alpha\leq1$.
These functions look actually much simpler when plotting them and I could determined thanks to a previous post that:
$$ \mathfrak{f}_1(\alpha )=\alpha^2 \left(\frac{1}{4}-\frac{2}{\pi^2}\right)-\frac{\alpha}{8}+\frac{1}{12} $$
Do you have any idea if it is possible to obtain the polynomial form of $\mathfrak{f}_2(\alpha )$ from $\mathfrak{f}_1(\alpha )$ since they look very close to each other? This would avoid an awfully complicated calculations. Thank you so much!