The expansion in ultraspherical polynomials: a simple procedure for the   fast computation of the ultraspherical coefficients
  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. 
Enrico De Micheli, Giovanni Alberto Viano
math.NA math-ph math.MP