On the Fourier coefficients of the derivative with respect to celebrated orthogonal systems

The main goal of this paper is to find the coefficients of the Jacobi polynomials and the integrals of Legendre polynomials expansion of the derivative of a function in terms of the coefficients in the expansion of the original function. More precisely, if {Qₙ} is a sequence or orthogonal polynomials, and if p(x)=∑ⱼ₌₀ⁿaⱼQₙ₋ⱼ(x) is such that p′(x)=∑ⱼ₌₀ⁿ⁻¹dⱼQₙ₋ⱼ₋₁(x), we find an explicit relation for the coefficients dⱼ, as linear combinations of the coefficients aⱼ. This will be done for two celebrated classes of orthogonal functions, namely the Jacobi polynomials and the integrals of the Legendre polynomials.