Legendre polynomials

Legendre polynomials
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Name	Legendre polynomials
Notation	P_n(x)
Type	Orthogonal polynomials
Domain	[−1, 1]
Parameter	degree n ∈ ℕ_0
Recurrence	three-term recurrence
Related	Rodrigues' formula, spherical harmonics, Sturm–Liouville theory

Legendre polynomials are a classical family of orthogonal polynomials defined on the interval [−1, 1] with deep connections to mathematical physics, spectral theory, and approximation theory. Developed in the context of potential theory and celestial mechanics by Adrien-Marie Legendre, they appear in solutions of Laplace's equation in spherical coordinates and in Gaussian quadrature. Their algebraic, analytic, and computational properties connect to Sturm–Liouville problems, special functions, and representation theory.

Definition and basic properties

The sequence P_n(x) is defined for nonnegative integer degree n as a polynomial of degree n with leading coefficient 2^n n! / (2n)!, normalized so that P_n(1)=1, and satisfying the Legendre differential equation, a second-order linear ordinary differential equation. Important algebraic properties include parity P_n(−x) = (−1)^n P_n(x), real roots that are simple and lie in (−1,1), and interlacing of zeros between successive degrees, features shared with families studied by Sturm, Hermite, and Chebyshev. Analytic continuation of P_n(x) connects to hypergeometric functions and to work by Gauss, Euler, and Riemann on differential equations.

Generating function and recurrence relations

A compact generating function encodes all degrees: the formal power series representation yields coefficients P_n(x) via expansion, a technique familiar from Eulerian generating functions and from methods used by Fourier, Cauchy, and Dirichlet. From the generating function one derives the standard three-term recurrence relation relating P_{n+1}, P_n, and P_{n−1}, which is central in algorithms by Golub and Welsch for computing Gaussian quadrature nodes associated with Legendre polynomials. Recurrence relations also tie into matrix models and spectral methods used in works by Lanczos and Trefethen on numerical linear algebra.

Orthogonality and normalization

Legendre polynomials form an orthogonal basis for L^2([−1,1]) with respect to the Lebesgue weight 1, a property exploited in expansions analogous to Fourier series in analyses by Parseval and Plancherel. The orthogonality relation ∫_{−1}^1 P_m(x) P_n(x) dx = 0 for m ≠ n and normalization constants allow construction of orthonormal sets used in Galerkin and spectral methods stemming from contributions by Ritz, Rayleigh, and Galerkin. Orthogonality underlies Gaussian quadrature of Gauss–Legendre type, as developed by Gauss and refined in numerical practice by Clenshaw and Curtis.

Rodrigues' formula and explicit expressions

Rodrigues' formula gives P_n(x) as (1/(2^n n!)) d^n/dx^n[(x^2−1)^n], an explicit differential representation paralleling Rodrigues' approach for classical orthogonal polynomials including Hermite and Laguerre. Closed-form expressions involve hypergeometric series {}_2F_1 and factorial coefficients appearing in combinatorial identities studied by Pascal, Vandermonde, and Newton. Explicit sums, derivative identities, and connection formulas link to the work of Jacobi on orthogonal polynomials and to Mehler and Heine asymptotic expansions used in the study of large-degree behavior.

Applications in physics and numerical methods

Legendre polynomials arise in solutions of Laplace's equation in spherical geometries encountered by Poisson, Laplace, and Kelvin, and in multipole expansions used in classical electrodynamics as developed by Coulomb, Maxwell, and Hertz. In quantum mechanics they appear in the separation of variables for the Schrödinger equation in central potentials, with seminal applications by Schrödinger, Pauli, and Dirac. Numerical methods leveraging Legendre bases include spectral methods and pseudospectral collocation advanced by Boyd and Canuto, and Gaussian quadrature rules central to approximation theory employed by Clenshaw, Curtis, and Golub. Engineering and geophysics applications draw on expansions used in seismology by Richter and in geodesy by Helmert and Gauss.

Associated Legendre functions and spherical harmonics

Associated Legendre functions P_n^m(x) generalize P_n(x) by introducing order m and appear in the construction of spherical harmonics Y_n^m(θ,φ), which form a complete orthonormal set on the sphere used extensively in the studies of Laplace, Dirichlet, and Fourier on manifolds. Spherical harmonics underpin representations in angular momentum theory developed by Wigner and Clebsch–Gordan, and they are central to scattering theory by Born and approximation techniques in computer graphics by Phong and Blinn. The ladder relations, orthogonality on the sphere, and addition theorems connect to group-theoretic approaches by Lie, Weyl, and Cartan, and to practical implementations in fast multipole methods by Greengard.

Adrien-Marie Legendre Legendre's work Sturm–Liouville theory Gauss Euler Riemann Jacobi Helmholtz Laplace Poisson Maxwell Schrödinger Dirac Wigner Clebsch–Gordan Golub Welsch Clenshaw Curtis Lanczos Trefethen Boyd Canuto Greengard Coulomb Hertz Rayleigh Galerkin Ritz Vandermonde Newton Pascal Mehler Heine Born Phong Blinn Helmert Euler–Maclaurin Fourier Euler–Lagrange Lie Weyl Cartan Gauss–Legendre quadrature Fast multipole method Spherical harmonics Hypergeometric function Stieltjes Orthogonal polynomials Spectral method Pseudospectral method Multipole expansion Angular momentum (quantum mechanics) Potential theory Celestial mechanics Seismology Geodesy Numerical linear algebra Approximation theory Special functions Hermite polynomials Laguerre polynomials Chebyshev polynomials Gaussian quadrature Rodrigues' formula Three-term recurrence Asymptotic analysis Representation theory

Category:Orthogonal polynomials