Explicit formulae for derivatives and primitives of orthogonal polynomials

TL;DR

This paper derives explicit formulas for derivatives and integrals of orthogonal polynomials, enabling efficient computation of their coefficients in generalized Fourier series, with applications demonstrated through tests.

Contribution

It provides new explicit formulas for derivatives and primitives of orthogonal polynomials based on their three-term recurrence relations.

Findings

Explicit formulas for matrix elements of integro-differential operators

Exact coefficients for specific orthogonal polynomial families

Demonstrated robustness through numerical tests

Abstract

In this work we deduce explicit formulae for the elements of the matrices that represent the action of integro-differential operators over the coefficients of generalized Fourier series. Our formulae are obtained by performing operations on the bases of orthogonal polynomials and result directly from the three-term recurrence relation satisfied by the polynomials. Moreover we give exact formulae for the coefficients for some families of orthogonal polynomials. Some tests are given to demonstrate the robustness of the formulas presented.