a generalized Fourier series
The ordinary Fourier series is a famous discovery: any reasonable periodic signal can be written as a sum of sines and cosines. But sines and cosines are special only because they are the eigenfunctions of one particular boundary value problem. A generalized Fourier series is the realisation that the same trick works with any complete, orthogonal family of eigenfunctions — not just sines and cosines, but Bessel functions, Legendre polynomials, and the rest.
If {y_n} is the orthogonal family of eigenfunctions of a Sturm-Liouville problem with weight w, then the generalized Fourier series of a function f is f(x) = sum of c_n y_n(x), with the generalized Fourier coefficients c_n = (integral of f y_n w dx) / (integral of y_n^2 w dx). It is exactly an eigenfunction expansion, but the name 'generalized Fourier series' emphasises the parallel with the classical case and the fact that all the familiar Fourier machinery — coefficient formulas, Parseval's energy identity, mean-square convergence — carries over verbatim once you replace sines and cosines by y_n and weight everything by w.
The point of the generalisation is that the right building blocks depend on the geometry of the problem. Sines and cosines are natural on an interval or a rectangle; on a disk the natural modes are Bessel functions, on a sphere they are Legendre polynomials and spherical harmonics. In each setting the corresponding generalized Fourier series is the language in which solutions of partial differential equations are written. Classical Fourier series is then just the most familiar member of a large, unified family, all flowing from Sturm-Liouville theory.
A Fourier-Bessel series expands a function on a disk of radius 1 in the radial eigenfunctions J_0(alpha_n r), where alpha_n are the zeros of the Bessel function J_0. The weight is w(r) = r, so c_n = (integral from 0 to 1 of f(r) J_0(alpha_n r) r dr) / (integral from 0 to 1 of J_0(alpha_n r)^2 r dr) — the same structure as a Fourier sine series, with J_0 in place of sin and weight r in place of 1.
Same coefficient machinery as classical Fourier series, but with problem-specific eigenfunctions and weight — here Bessel functions weighted by r.
Classical sine and cosine Fourier series are not a separate theory but the simplest special case (p = 1, q = 0, w = 1). Recognising this prevents the common error of thinking Fourier coefficient formulas are unique to trig functions — they are the same formula with weight 1.