Complete bases of canonical polynomials and the full recursive formulation of the spectral-tau method

Abstract: A decomposition of a higher order linear differential operator with polynomial coefficients into a direct sum of two factor operators is obtained. This leads to a lower echelon matrix representation for operators of the above mentioned type arising from a finite sequence of elementary operations. As a consequence, a complete basis of canonical polynomials is generated via a modified Ortiz' recurrence formula. The approach presented in this paper enables us to curry out the full and self-operative recursive implementation of the spectral-tau method. These results can be applied to a variety of operator equations associated with infinite matrices in lower row echelon form, having implications for a broad family of discrete and continuous numerical methods, interpreted as special realizations of the tau method under specific perturbation terms.