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This paper deals with the analysis of the orthogonality of a monic polynomial sequencedefined as a linear combination of a sequence of monic orthogonal polynomialswithwherefor. Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials. We obtain the connection between the Jacobi matrices associated withand, respectively, by using an LU factorization. Some special cases of the above type relation are analysed. [ABSTRACT FROM AUTHOR]