= Legendre expansion of the sign function
{c}
{title2=$\langle\operatorname{sgn},p_{2m+1}\rangle=(-1)^m\sqrt{(4m+3)/2}\,(2m)!/[2^{2m}m!(m+1)!]$}
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 $\int_0^1P_n=P_n'(0)/(n(n+1))$. 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 $\sum_{m\geq0}((2m)!)^2(4m+3)/[2^{4m}(m!(m+1)!)^2]=4$.
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