Spectral Green function of a regular Sturm-Liouville problem ($G_{\mu }(x,\xi )={\sum }_n{y}_n(x)y_n(\xi )/({\lambda }_n-\mu )$)

Normalize the real eigenfunctions using $\int r{y}_m{y}_n={\delta }_{mn}$. For $\mu$ outside the spectrum, applying the operator to the displayed Green function expansion gives $r(x){\sum }_n{y}_n(x)y_n(\xi )=\delta (x-\xi )$ by weighted completeness. This is the kernel with respect to ordinary source integration $d\xi$; using the weighted measure would change its normalization.

For $L=d^2/d{x}^2$ 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.