Explicit formulas for the Fourier coefficients of the Legendre polynomials can be found in the Bateman Manuscript Project. However, similar formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials.
N. Greene, “Fourier Series of Orthogonal Polynomials,” in: Recent Advances on Applied Mathematics, Proceedings of the American Conference on Applied Mathematics, Cambridge, Massachusetts, March 24-26, 2008, pp. 247-251.