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.

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.

Given {Pn} n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,Qn(x) = Pn(x) + a1Pn−1(x) + · · · + a k P n−k , a k = 0, n > k. Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn} n≥0 as well as an interesting interpretation in terms of the Jacobi matrices associated with {Pn} n≥0 and {Qn} n≥0 .Moreover, in the case k = 2, we characterize the families {Pn} n≥0 such that the corresponding polynomials {Qn} n≥0 are also orthogonal.

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.