Orthogonal polynomials associated with an inverse spectral transform. The cubic case

The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn}n ? o defined by Qn(X) = Pn(X) + SnPn-1(X) + tnPn-2(X) + rnPn-3(X), n ? 1, where rn ? 0, n ? 3, and {Pn}n?0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters rn, sn, and tn can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.