Associated Legendre polynomials

where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively.

When m is zero and ℓ integer, these functions are identical to the Legendre polynomials.

The Legendre ordinary differential equation is frequently encountered in physics and other technical fields.

Associated Legendre polynomials play a vital role in the definition of spherical harmonics.

, where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms of derivatives of ordinary Legendre polynomials (m ≥ 0)

Indeed, equate the coefficients of equal powers on the left and right hand side of

with simple monomials and the generalized form of the binomial coefficient.

Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:

Also, they satisfy the orthogonality condition for fixed ℓ:

The differential equation is clearly invariant under a change in sign of m. The functions for negative m were shown above to be proportional to those of positive m:

This definition also makes the various recurrence formulas work for positive or negative m.)

For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed.

{\displaystyle {\begin{aligned}{\frac {1}{2}}\int _{-1}^{1}P_{l}^{u}(x)P_{m}^{v}(x)P_{n}^{w}(x)dx={}&{}(-1)^{s-m-w}{\frac {(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!

These functions may actually be defined for general complex parameters and argument:

These functions are most useful when the argument is reparameterized in terms of angles, letting

In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved.

Together, they make a set of functions called spherical harmonics.

These functions express the symmetry of the two-sphere under the action of the Lie group SO(3).

[citation needed] What makes these functions useful is that they are central to the solution of the equation

In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is

is solved by the method of separation of variables, one gets a φ-dependent part

They are all orthogonal in both ℓ and m when integrated over the surface of the sphere.

are the spherical harmonics, and the quantity in the square root is a normalizing factor.

Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identity[5]

Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).

When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically of the form

The Legendre polynomials are closely related to hypergeometric series.

In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3).

There are many other Lie groups besides SO(3), and analogous generalizations of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces.

Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.