Jacobi quadrature | Mehler quadrature

Jacobi-Gauss quadrature, also called Jacobi quadrature or Mehler quadrature, is a Gaussian quadrature over the interval [-1, 1] with weighting function W(x) = (1 - x)^α (1 + x)^β. The abscissas for quadrature order n are given by the roots of the Jacobi polynomials P_n^(α, β)(x). The weights are w_i | = | -(A_(n + 1) γ_n)/(A_n P_n^(α, β)'(x_i) P_(n + 1)^(α, β)(x_i)) | = | A_n/A_(n - 1) γ_(n - 1)/(P_(n - 1)^(α, β)(x_i) P_n^(α, β)'(x_i)), where A_n is the coefficient of x^n in P_n^(α, β)(x).

Carl Friedrich Gauss | Carl Jacobi