An explicit formula for the Fourier coefficient of the Legendre polynomials can be found in the Bateman Manuscript Project. However, 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. The methods described here apply in principle to a class of polynomials, including non-orthogonal polynomials.
N. Greene, “Formulas for the Fourier Series Coefficients of Orthogonal Polynomials in Terms of Special Functions,” International Journal of Mathematical Models and Methods in Applied Sciences, Vol. 2, Issue 3, 2008, pp. 317-320.