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Numerical Algorithms

, Volume 60, Issue 1, pp 51–73 | Cite as

Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

  • Herbert Dueñas
  • Edmundo J. Huertas
  • Francisco Marcellán
Original Paper

Abstract

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product
$$ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1 $$
where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler–Heine formula as well as a Plancherel–Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N.

Keywords

Orthogonal polynomials Laguerre polynomials Laguerre–Sobolev-type orthogonal polynomials Laguerre polynomials Bessel functions Rescaled polynomials Asymptotics Plancherel–Rotach type formula Outer relative asymptotics Mehler–Heine type formula 

Mathematics Subject Classification (2010)

33C47 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Herbert Dueñas
    • 1
  • Edmundo J. Huertas
    • 2
  • Francisco Marcellán
    • 2
  1. 1.Departamento de Matemáticas Ciudad UniversitariaUniversidad Nacional de ColombiaBogotáColombia
  2. 2.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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