Three 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 is motivated by the role they play in certain finite difference schemes used in the discretization of the angular Fokker-Planck diffusion operator.

翻译：提出了Legendre节点与权重之间的三种不同关系，与Gauss-Legendre求积的圆定理和梯形定理不同，这些关系在正交区间$(-1,1)$上一致成立。这些性质有强渐近证据支持。这些结果的研究受到它们在离散角向Fokker-Planck扩散算子的某些有限差分格式中所起作用的启发。