  We present a simple and fast algorithm for the computation of the
coefficients of the expansion of a function f(cos u) in ultraspherical
(Gegenbauer) polynomials. We prove that these coefficients coincide with the
Fourier coefficients of an Abel-type transform of the function f(cos u). This
allows us to fully exploit the computational efficiency of the Fast Fourier
Transform, computing the first N ultraspherical coefficients in just O (N log_2
N) operations.
