Legendre Polynomials

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Legendre polynomials

[lə′zhän·drə ‚päl·i′nō·mē·əlz]
A collection of orthogonal polynomials which provide solutions to the Legendre equation for nonnegative integral values of the parameter.

Legendre Polynomials


a system of polynomials of successively increasing degree. The polynomials were first investigated by A. Legendre and P. Laplace independently of each other between 1782 and 1785. For n = 0, 1, 2, …, the Legendre polynomials Pn(x) can be defined by the formula

The first few polynomials are

p0 (x) = 1

p1 (x) = x

p2 (x) = 1/2(3x2 - 1)

p3 (x) = 1/2(5x2 - 3x)

p4 (x) = 1/8(35x4 - 30x2 + 3)

p5 (x) = 1/8(63x5 - 70x3 + 15x)

All the zeros of P n (x) are real, lie in the interval [–1, + 1], and alternate with the zeros of Pn + 1 (x). The Legendre polynomials are a complete set of orthogonal polynomials on the interval [–1, + 1]. Thus, it is possible to expand an arbitrary function /(jc) integrable over the interval [– 1, +1] in a series of Legendre polynomials:


The type of convergence of this series is roughly the same as that of a Fourier series. The Legendre polynomials are given explicitly by the formula

The generating function is

that is, the Legendre polynomials are the coefficients in the expansion of this function in powers of t. They are recursively defined by

nPn (x) + (n - 1)Pn-2 (x) - (2n - 1)xPn-1 (x) = 0

Pn (x)satisfies the differential equation

which arises when separating the variables in Laplace’s equation in spherical coordinates.


Janke, E., F. Emde, and F. Lösch. Spetsial’nye funktsii; grafiki, tablitsy, 2nd ed. Moscow, 1968. (Translated from German.)
Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. MoscowLeningrad, 1963.


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Set the Legendre-Gauss points [mathematical expression not reproducible] as the zeros of the Legendre polynomials [L.sub.N+1](t).
where k =1, 2, ..., [??] = 2n - 1, n =1, 2, ..., [2.sup.k-1], m = 0, 1, ..., M - 1 is the degree of the Legendre polynomials and M is a fixed positive integer, and [P.sub.m](t) are the Legendre polynomials of degree m.