the Legendre polynomials
/ luh-ZHAHND-ruh /
When a physics problem has spherical symmetry -- the field of a charge distribution, the temperature of a ball -- the natural building blocks for the angular direction are the Legendre polynomials: a family P_0, P_1, P_2, ... of simple polynomials in cos(theta) that are perfectly 'independent' of one another in a precise averaging sense. Any reasonable function of the polar angle is a weighted sum of them.
P_n(x) is a polynomial of degree n on the interval [-1, 1] (with x = cos theta), the solution of Legendre's equation d/dx[(1 - x^2) dP/dx] + n(n+1) P = 0. The first few are P_0 = 1, P_1 = x, P_2 = (3x^2 - 1)/2. They are orthogonal: the integral from -1 to 1 of P_m(x) P_n(x) dx = 0 for m not equal to n, and equals 2/(2n + 1) when m = n. Rodrigues' formula generates them all: P_n(x) = (1/(2^n n!)) (d/dx)^n (x^2 - 1)^n, and they are the coefficients in the generating function 1/sqrt(1 - 2 x t + t^2) = sum of P_n(x) t^n.
That generating function is literally the multipole expansion: 1/|r - r'| expands in Legendre polynomials, giving the monopole, dipole, quadrupole, ... terms of a potential. Legendre's equation is exactly the polar-angle piece you get when you separate Laplace's equation in spherical coordinates, and multiplying by the azimuthal factor promotes them to the spherical harmonics. They are the archetype of a complete orthogonal set, the finite-interval analogue of the sines and cosines of Fourier series.
The potential of an off-axis point charge expands as 1/|r - r'| = (1/r) times the sum of (r'/r)^n P_n(cos theta) for r greater than r'. The n = 0 term is the monopole 1/r; n = 1 is the dipole; the P_n are the multipoles.
Legendre polynomials are the multipole coefficients of a spherical potential.
Orthogonality holds only with the flat weight 1 on the interval [-1, 1]; change the interval or the weight and you get a different family (associated Legendre, Chebyshev, ...). The weight is part of the definition.