Bessel functions, Legendre polynomials and other special functions

Ordinary Fourier series work because sines and cosines are the eigenfunctions of the simplest second-order operator on a flat, periodic domain. Change the domain's shape — a cylinder instead of a line, a sphere instead of a plane — and separating variables in Laplace's or the wave equation still leaves behind a second-order ODE for the "radial" or "angular" part, but it's no longer one that sine and cosine solve. Each geometry has its own eigenfunctions, and mathematicians named the families after whoever worked them out: Bessel functions for cylindrical symmetry, Legendre polynomials for spherical.

The same idea reaches further than pure geometry. The quantum harmonic oscillator's Schrödinger equation reduces, after stripping off a Gaussian factor, to the ODE solved by Hermite polynomials. The hydrogen atom's radial equation reduces to the one solved by (associated) Laguerre polynomials. In every case the pattern is identical: a physical symmetry forces a specific ODE, and a specific named family of functions is simply what "the well-behaved solutions to that ODE" are called.

What makes these families more than a bestiary of formulas is that each one is a complete orthogonal set on its natural domain — exactly like sines and cosines are complete on an interval. That means any reasonably well-behaved function on a disk can be expanded as a sum of Bessel functions, any function on a sphere as a sum of Legendre polynomials (or, allowing for the azimuthal angle too, spherical harmonics), the same way an arbitrary periodic signal is expanded as a Fourier series. Recognising which family a problem calls for is really just recognising its symmetry.