Δ-Sobolev orthogonal polynomials of Meixner type: asymptotics and limit relation

  • Authors:

  • I. Area;E. Godoy;F. Marcellán;J. J. Moreno-Balcázar

  • Affiliations:

  • Departamento de Matemática Aplicada II, E.T.S.E. de Telecomunicación, Universidade de Vigo, Campus Lagoas-Marcosende, 36200 Vigo, Spain;Departamento de Matemática Aplicada II, E.T.S.I. Industriales, Universidade de Vigo, Lagoas-Marcosende, 36200 Vigo, Spain;Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III, Avenida de la Universidad, 30, 28911 Leganés-Madrid, Spain;Departamento de Estadística y Matemática Aplicada, Edificio Científico Técnico III, Univ. de Almería Instituto Carlos I de Física Teórica y Computacional, Univ. ...

  • Venue:
  • Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
  • Year:
  • 2005

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Abstract

Let {Qn(x)}n be the sequence of monic polynomials orthogonal with respect to the Sobolev-type inner product 〈p(x), r(x)〉S = 〈u0, p(x)r(x)〉 + λ〈u1, (Δp)(x)(Δr)(x)〉, where λ ≥ 0, (Δf)(x) = f(x+1) - f(x) denotes the forward difference operator and (u0, u1) is a Δ-coherent pair of positive-definite linear functionals being u1 the Meixner linear functional. In this paper, relative asymptotics for the {Qn(x)}n sequence with respect to Meixner polynomials on compact subsets of C\[0, + ∞) is obtained. This relative asymptotics is also given for the scaled polynomials. In both cases, we deduce the same asymptotics as we have for the self-Δ-coherent pair, that is, when u0 = u1 is the Meixner linear functional. Furthermore, we establish a limit relation between these orthogonal polynomials and the Laguerre-Sobolev orthogonal polynomials which is analogous to the one existing between Meixner and Laguerre polynomials in the Askey scheme.