LegendrePolynome

LegendrePolynome, commonly called Legendre polynomials, are a sequence of orthogonal polynomials Pn(x) defined on the interval [-1, 1] with respect to the unit weight. They arise in problems with spherical symmetry, notably in solving Laplace’s equation in spherical coordinates and in expanding functions on the sphere.

Definition and generating tools: Legendre polynomials can be defined by Rodrigues’ formula, Pn(x) = (1/(2^n n!)) d^n/dx^n [(x^2 - 1)^n]. They also have a generating function 1/√(1 - 2xt + t^2) = ∑_{n=0}^∞ Pn(x) t^n for |t| < 1, which encodes all polynomials in the family.

Key properties: The polynomials are orthogonal on [-1, 1], satisfying ∫_{-1}^1 Pn(x) Pm(x) dx = 2/(2n+1) δ_nm. They solve the Legendre differential equation (1 - x^2) y'' - 2x y' + n(n+1) y = 0. A standard recurrence relates consecutive polynomials: (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x), with P_0(x) = 1 and P_1(x) = x. Parity is given by P_n(-x) = (-1)^n P_n(x).

Examples and applications: Early polynomials include P_0(x) = 1, P_1(x) = x, P_2(x) = (3x^2 - 1)/2, and P_3(x) = (5x^3 - 3x)/2. Legendre polynomials are used in spherical harmonics, potential theory, and numerical integration via Gauss-Legendre quadrature. They also connect to associated Legendre functions P_n^m(x) for m > 0 and to various normalization conventions in physics and engineering.