# Legendre polynomials

spherical polynomials

Polynomials orthogonal on the interval with unit weight . The standardized Legendre polynomials are defined by the Rodrigues formula and have the representation The formulas most commonly used are:    The Legendre polynomials can be defined as the coefficients in the expansion of the generating function where the series on the right-hand side converges for .

The first few standardized Legendre polynomials have the form    The Legendre polynomial of order satisfies the differential equation (Legendre equation) which occurs in the solution of the Laplace equation in spherical coordinates by the method of separation of variables. The orthogonal Legendre polynomials have the form and satisfy the uniform and weighted estimates  Fourier series in the Legendre polynomials inside the interval are analogous to trigonometric Fourier series (cf. also Fourier series in orthogonal polynomials); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function at a point converges if and only if the trigonometric Fourier series of the function converges at the point . In a neighbourhood of the end points the situation is different, since the sequence increases with speed . If is continuous on and satisfies a Lipschitz condition of order , then the Fourier–Legendre series converges to uniformly on the whole interval . If , then this series generally diverges at the points .

These polynomials were introduced by A.M. Legendre .