Legendre expansion of the sign function ($⟨\mathrm{sgn},p_{2m+1}⟩=(-1{)}^m\sqrt{(4m+3)/2}(2m)!/[2^{2m}m!(m+1)!]$)

For normalized Legendre polynomials $p_n=\sqrt{(2n+1)/2}{P}_n$, parity makes the sign function orthogonal to every even index. Integrating the Legendre differential equation gives ${\int }_0^1{P}_n=P_n^{\prime }(0)/(n(n+1))$. The differentiated generating function $z(1+z^2{)}^{-3/2}$ gives the derivatives at zero and hence the displayed odd coefficients. The Parseval identity and squared norm two then evaluate ${\sum }_{m\ge 0}((2m)!{)}^2(4m+3)/[2^{4m}(m!(m+1)!{)}^2]=4$.