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Monotonicity and asymptotics of zeros of Laguerre-Sobolev-type orthogonal polynomials of higher order derivatives

Authors: Francisco Marcellán and Fernando R. Rafaeli
Journal: Proc. Amer. Math. Soc. 139 (2011), 3929-3936
MSC (2010): Primary 42C05; Secondary 33C47
Published electronically: March 10, 2011
MathSciNet review: 2823039
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Abstract: In this paper we analyze the location of the zeros of polynomials orthogonal with respect to the inner product \begin{equation} \langle p,q\rangle = \displaystyle \int _{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{(j)}(0)q^{(j)}(0), \end{equation} where $\alpha >-1$, $N\geq 0,$ and $j\in \mathbb {N}.$ In particular, we focus our attention on their interlacing properties with respect to the zeros of Laguerre polynomials as well as on the monotonicity of each individual zero in terms of the mass $N.$ Finally, we give necessary and sufficient conditions in terms of $N$ in order for the least zero of any Laguerre-Sobolev-type orthogonal polynomial to be negative.

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Additional Information

Francisco Marcellán
Affiliation: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, 28911 Leganés, Spain

Fernando R. Rafaeli
Affiliation: Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, São Paulo, Brazil
Address at time of publication: Departamento de Matemática, Estatística e Computação/FCT, Universidade Estadual Paulista-UNESP, 19060-900 Presidente Prudente, São Paulo, Brazil

Keywords: Laguerre orthogonal polynomials, Laguerre-Sobolev-type orthogonal polynomials, zeros, interlacing, monotonicity, asymptotics.
Received by editor(s): November 25, 2009
Received by editor(s) in revised form: September 7, 2010
Published electronically: March 10, 2011
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.