Fourier Series of Orthogonal Polynomials

It follows from Bateman [4] page 213 after setting = 1 2 . It can also be found with slight modi cation in Bateman [5] page122. However we are not aware of any reference where explicit formulas for the Fourier coef cients for Gegenbauer, Jacobi, Laguerre and Hermite polynomials can be found. In this article we use known formulas for the connection coef cients relating an arbitrary orthogonal polynomial to the Legendre polynomials to derive explicit formulas. Although we detail the formulas for the classical orthogonal polynomials, the method can be used to write explicit Fourier coef cients for any class of polynomials. The formulas were developed by this author, Greene [11], in studying the Gegenbauer reconstruction method of Gottlieb and Shu [7, 8, 9, 10], which is a technique for overcoming the spurious oscillations known as the Gibbs phenomenon which occur in Fourier and orthogonal polynomial approximations to piecewise smooth functions.