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 [1].

See also the references to Orthogonal polynomials.

References

[1]	A.M. Legendre, Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants , 10 (1785) pp. 411–434
[2]	E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931)

Comments

Legendre polynomials belong to the families of Gegenbauer polynomials; Jacobi polynomials and classical orthogonal polynomials. They can be written as hypergeometric functions (cf. Hypergeometric function). Their group-theoretic interpretation as zonal spherical functions on the two-dimensional sphere serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.