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.

Let (Pn)n and (Qn)n be two sequences of monic polynomials linked by a type structure relation such aswhere (rn)n, (sn)n and (tn)n are sequences of complex numbers.Firstly, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences (Pn)n and (Qn)n are orthogonal with respect to regular moment linear functionals u and v, respectively.Secondly, assuming that the above relation is non-degenerate and (Pn)n is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence (Qn)n in terms of the coefficients of the polynomials Φ and Ψ which appear in the rational transformation (in the distributional sense) Φu = Ψv .Some illustrative examples of the developed theory are presented.2000 Mathematics Subject Classification. 42C05, 33C45.

Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 = \mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.Comment: 28 pages. To appear in Numerical Algorithm

C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre-Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán. 2000MSC: 42C05