Past exam of the analysis course of the University of Oxford / 2009 / B / 4 / ii

The generating function gives $P_n(-t)=(-1{)}^n{P}_n(t)$, so the odd function $f$ is orthogonal to every even-degree Legendre polynomial. For $n\ge 1$, integrate the self-adjoint form of the Legendre differential equation from zero to one:Thus ${\int }_0^1{P}_n=P_n^{\prime }(0)/(n(n+1))$. Differentiating the generating function in $t$ and then setting $t=0$ givesCombining these identities with the normalization of $p_n$ yieldsFor the Legendre expansion of the sign function, $\Vert f{\Vert }_2^2=2$, so the Parseval identity givesTherefore the requested sum equals