Legendre polynomials

Summary

The Legendre polynomials $P_0, P_1, P_2, \dots$ are the polynomials obtained by making $1, x, x^2, \dots$ orthogonal on $[-1, 1]$ with the constant weight, normalised by $P_n(1) = 1$. They are the simplest classical orthogonal polynomials, the Jacobi case $\alpha = \beta = 0$. They solve Legendre's equation and appear in the Laplace equation in spherical coordinates, in multipole expansions, and in Gauss–Legendre quadrature.