Some novel results on Boubaker polynomials leading to an efficient orthogonalization

The orthogonal polynomials have been recognized in literature due to their compatibility in providing robust and accurate solutions of different nonlinear, singular, and complex problems in science and engineering. The Boubaker polynomial (BPs) and its variants to date have been at galore in the recent literature in this context. Although exhaustive applications of the BPs and its variants have been performed, systematic orthogonalization of the polynomials is still a persistent problem. In one way or another, the existing variants compromise in being an efficient orthogonalization due to inclusion of imaginary zeros, rounding off errors of coefficients, the choice of appropriate weight function, and symmetry with the conventional BPs. In this study, we derive a class of weight-functions in a Hilbert space which accommodates an efficient orthogonalization of the BPs for the first time. We also prove theorems on the consequent recurrence, orthogonality and orthonormality relations for the proposed orthogonal BPs (POBPs). The characteristic differential equation and its spectral form have also been derived. The results of this study are a basis for the applicability of POBPs where existing attempts suffered due to lack of efficient orthogonalization.