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Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

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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.

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Correspondence to Edmundo J. Huertas.

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The work of the second and third authors has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01.

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Dueñas, H., Huertas, E.J. & Marcellán, F. Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials. Numer Algor 60, 51–73 (2012). https://doi.org/10.1007/s11075-011-9511-4

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