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

## DOI:

https://doi.org/10.62918/hjma.v2i1.19## Keywords:

orthogonal polynomials, Jacobi polynomials, special functions, Fourier coefficient, Legendre polynomials## Abstract

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_{n}} is a sequence or orthogonal polynomials, and if p(x)=∑_{j=0}ⁿa_{j}Q_{n-j}(x) is such that p′(x)=∑_{j=0}ⁿ⁻¹d_{j}Q_{n-j-1}(x), we find an explicit relation for the coefficients d_{j}, as linear combinations of the coefficients a_{j}. This will be done for two celebrated classes of orthogonal functions, namely the Jacobi polynomials and the integrals of the Legendre polynomials.

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*Hilbert Journal of Mathematical Analysis*,

*2*(1), 034–045. https://doi.org/10.62918/hjma.v2i1.19

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