A family of singular semi-classical functionals

Medem, J. C.

doi:10.1016/s0019-3577(02)80015-5

Cited by 9 publications

(9 citation statements)

References 3 publications

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“…must hold, where a and p are the leading coefficients of φ and ψ (resp.). J.C. Medem [18] found a regular functional u and a pair of non-admissible polynomials (φ, ψ) satisfying (4). For this reason we dropped the admissibility condition in the above definition of semiclassical functional.…”

Section: Remark 21mentioning

confidence: 99%

On linearly related sequences of derivatives of orthogonal polynomials

Jesus

Petronilho

2008

Journal of Mathematical Analysis and Applications

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We discuss an inverse problem in the theory of (standard) orthogonal polynomials involving two orthogonal polynomial families (P n ) n and (Q n ) n whose derivatives of higher orders m and k (resp.) are connected by a linear algebraic structure relation such asfor all n = 0, 1, 2, . . . , where M and N are fixed nonnegative integer numbers, and r i,n and s i,n are given complex parameters satisfying some natural conditions. Let u and v be the moment regular functionals associated with (P n ) n and (Q n ) n (resp.). Assuming 0 m k, we prove the existence of four polynomials Φ M+m+i and Ψ N+k+i , of degrees M + m + i and N + k + i (resp.), such thatthe (k − m)th-derivative, as well as the left-product of a functional by a polynomial, being defined in the usual sense of the theory of distributions. If k = m, then u and v are connected by a rational modification. If k = m + 1, then both u and v are semiclassical linear functionals, which are also connected by a rational modification. When k > m, the Stieltjes transform associated with u satisfies a non-homogeneous linear ordinary differential equation of order k − m with polynomial coefficients.

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Section: Remark 21mentioning

confidence: 99%

On linearly related sequences of derivatives of orthogonal polynomials

Jesus

Petronilho

2008

Journal of Mathematical Analysis and Applications

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“…The last condition is usually called the admissibility condition (for a detailed study of this condition see [21,22] and references therein). That this condition was necessary was established in [23].…”

Section: The Q-linear Lattices: the "Q-hahn Tableau"mentioning

confidence: 99%

On characterizations of classical polynomials

Álvarez-Nodarse¹

2006

Journal of Computational and Applied Mathematics

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It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients, Rodrigues formula, etc. In this paper we present a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the "q-Hahn tableau". Finally, a detailed discussion of a characterization by Marcellán et al. is presented.

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“…Furthermore, in the classical case we can give nonlinear recurrence relations for the square of the norms of the Sobolev polynomials and for the coefficients e n (λ) appearing in (17).…”

Section: Definition 24mentioning

confidence: 99%

The semiclassical Sobolev orthogonal polynomials: A general approach

Costas-Santos¹,

Moreno-Balcázar²

2011

Journal of Approximation Theory

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We say that the polynomial sequence (Q (λ) n ) is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product ⟨ p, r ⟩ S = ⟨u, p r ⟩ + λ ⟨u, D p Dr ⟩ , where u is a semiclassical linear functional, D is the differential, the difference or the q-difference operator, and λ is a positive constant.In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional u.The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator D considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.

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A family of singular semi-classical functionals

Cited by 9 publications

References 3 publications

On linearly related sequences of derivatives of orthogonal polynomials

On linearly related sequences of derivatives of orthogonal polynomials

On characterizations of classical polynomials

The semiclassical Sobolev orthogonal polynomials: A general approach

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