Uniform relations between the Gauss-Legendre nodes and weights

Four different relations between the Legendre nodes and weights are presented which, unlike the circle and trapezoid theorems for Gauss-Legendre quadrature, hold uniformly in the whole interval of orthogonality $(-1,1)$. These properties are supported by strong asymptotic evidence. The study of these results was originally motivated by the role some of them play in certain finite difference schemes used in the discretization of the angular Fokker-Planck diffusion operator.