the Legendre polynomials
/ luh-ZHAHN-druh /
These are the polynomials that play, on the sphere, the role sines and cosines play on a line. Whenever you separate Laplace's equation in spherical coordinates — gravitational potentials, electric fields, the shape of a planet, atomic orbitals — the angular part hands you the Legendre polynomials. They are the natural alphabet for functions that live on a sphere.
The Legendre polynomials P_0, P_1, P_2, ... are the eigenfunctions of the singular Sturm-Liouville problem -((1 - x^2) P')' = lambda P on [-1, 1], where x = cos(theta) is the cosine of the polar angle. The endpoints x = plus/minus 1 are singular because the coefficient 1 - x^2 vanishes there; demanding bounded solutions forces lambda = n(n+1), and then the bounded solution is a polynomial of degree n: P_0 = 1, P_1 = x, P_2 = (3 x^2 - 1)/2, and so on. They are orthogonal on [-1, 1] with the plain weight w = 1: integral from -1 to 1 of P_m P_n dx = 0 for m not n. Multiplying them by the azimuthal factors e^(i*m*phi) builds the spherical harmonics.
Their reach is enormous. The expansion of any function on a sphere in spherical harmonics — used in geodesy to model the Earth's gravity field, in cosmology to analyze the cosmic microwave background, in chemistry for atomic orbitals (the s, p, d shapes) — is a Legendre / spherical-harmonic series, the spherical version of a generalized Fourier series. The famous multipole expansion of a potential is exactly an expansion in Legendre polynomials of cos(theta).
A point charge off-center inside a grounded sphere is solved by expanding 1/|r - r'| in Legendre polynomials: 1/|r - r'| = sum over n of (r_small^n / r_big^(n+1)) P_n(cos theta), where theta is the angle between r and r'. Each P_n captures one 'multipole' — monopole, dipole, quadrupole — of the potential.
The multipole expansion of a potential is a Legendre series in cos(theta).
Watch the normalization conventions: P_n(1) = 1 is the standard choice, but the orthonormal version carries a factor sqrt((2n+1)/2), and physicists' spherical harmonics use yet other constants. Always check which convention a formula assumes before combining results from different sources.