Math.Comput.Sci. https://doi.org/10.1007/s11786-020-00465-1 Mathematics in Computer Science Explicit Formulae for Integro-Differential Operational Matrices José M. A. Matos · Maria João Rodrigues · João Carrilho de Matos Received: 11 July 2019 / Revised: 16 January 2020 / Accepted: 16 February 2020 © Springer Nature Switzerland AG 2020 Abstract In this work we deduce explicit formulae for the elements of the matrices representing 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. Keywords Integro-differential equations · Operational matrices · Orthogonal polynomials Mathematics Subject Classification 42C99 · 45A05 · 80M22 1 Introduction In some spectral methods the operational matrices that transform integro-differential problems into algebraic prob- lems are often obtained using a similarity transformation [14]. For high degree approximation the accuracy of the approximate solutions is degraded by the bad conditioning of the matrices involved. In recent works, dealing with the extension of spectral methods to systems of nonlinear integro-differential problems [21] and to problems with non-polynomial coefficients [22], the error propagation when working with operational matrices is referred as a drawback. This fact is of great importance when there is need of a large number of coefficients computed with great precision, as it is in the case with Frobenius-Padé approximation allowing the computation of rational approximants of series with unknown coefficients [11]. J. M. A. Matos · J. C. de Matos Laboratório de Engenharia Matemática, Instituto Superior de Engenharia do Porto, Porto, Portugal e-mail: jma@isep.ipp.pt J. C. de Matos e-mail: jem@isep.ipp.pt M. J. Rodrigues (B ) · J. M. A. Matos Centro de Matemática da, Universidade do Porto, Porto, Portugal e-mail: mjsrodri@fc.up.pt M. J. Rodrigues Faculdade de Ciências da, Universidade do Porto, Porto, Portugal