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 Q₍ is a sequence or orthogonal polynomials, and if p (x) =∑₉=₀ⁿa₉Q₍-₉ (x) is such that p′ (x) =∑₉=₀ⁿ⁻¹d₉Q₍-₉-₁ (x), we find an explicit relation for the coefficients d₉, as linear combinations of the coefficients a₉. This will be done for two celebrated classes of orthogonal functions, namely the Jacobi polynomials and the integrals of the Legendre polynomials.