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Let {P n } be a sequence of polynomials orthogonal with respect a linear functional u and {Q n } a sequence of polynomials defined byWe find necessary and sufficient conditions in order to {Q n } be a sequence of polynomials orthogonal with respect to a linear functional v. Furthermore we prove that the relation between these linear functionals is (x −ã)u = λ(x − a)v. Even more, if u and v are linked in this way we get that {P n } and {Q n } satisfy a formula as above.

Let u be a quasi-definite linear functional. We find necessary and sufficient conditions in order to the linear functional v satisfying (x −ã)u = λ(x − a)v be a quasi-definite one. Also we analyze some linear relations linking the polynomials orthogonal with respect to u and v.

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In this paper, we study orthogonal polynomials with respect to the bilinear formwhere u is a quasi-definite (or regular) linear functional on the linear space IP of real polynomials, c is a real number, AT is a positive integer number, A is a symmetric N x N real matrix such that each of its principal submatrices are regular, and F(c) = (/(c), /'(c),...,/^J V~1 H C ))> G(c) = (g(c)ig f (c),...,g( N~1 \c)). For these non-standard orthogonal polynomials, algebraic and differential properties are obtained, as well as their representation in terms of the standard orthogonal polynomials associated with u.

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Let {Pn} n≥0 be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional u and {Qn} n≥0 a sequence of polynomials defined byWe obtain a new characterization of the orthogonality of the sequence {Qn} n≥0 with respect to a linear functional v, in terms of the coefficients of a quadratic polynomial h such that h(x)v = u.We also study some cases in which the parameters sn and tn can be computed more easily, and give several examples.Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with {Pn} n≥0 and {Qn} n≥0 is presented.