Asymptotic properties of Laguerre---Sobolev type orthogonal polynomials

Authors:
Herbert Dueñas;Edmundo J. Huertas;Francisco Marcellán
Affiliations:
Departamento de Matemáticas Ciudad Universitaria, Universidad Nacional de Colombia, Bogotá, Colombia;Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain 28911;Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain 28911
Venue:
Numerical Algorithms
Year:
2012

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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驴驴驴驴驴+驴, and a驴驴驴驴驴驴驴. 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.