Odd reciprocal-square sum from a Dirichlet Green function
= Odd reciprocal-square sum from a Dirichlet Green function
{title2=$\sum_{j\ge0}(2j+1)^{-2}=\pi^2/8$}
For $L=d^2/dx^2$ on $(0,\pi)$ with zero endpoints, the zero-parameter <Green function> is $-x_<(\pi-x_>)/\pi$. Its <eigenfunctions> are $\sqrt{2/\pi}\sin(nx)$ with eigenvalues $-n^2$. At $x=\xi=\pi/2$, the kernel is $-\pi/4$ and its expansion contains precisely the odd reciprocal squares, giving the displayed identity.