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On Freud–Sobolev type orthogonal polynomials

  • Luis E. GarzaEmail author
  • Edmundo J. Huertas
  • Francisco Marcellán
Article
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Abstract

In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product
$$\begin{aligned} \left\langle p,q\right\rangle _{1}=\int _{\mathbb {R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{\prime }(0)q^{\prime }(0), \end{aligned}$$
where pq are polynomials, \(M_{0}\) and \(M_{1}\) are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as an application, an algorithm to compute their zeros is presented. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.

Keywords

Orthogonal polynomials Exponential weights Freud–Sobolev type orthogonal polynomials Zeros Interlacing Electrostatic interpretation 

Mathematics Subject Classification

Primary 33C45 Secondary 33C47 

Notes

Acknowledgements

We thank the anonymous referee for carefully reading the manuscript and for giving constructive comments, which substantially helped us to improve the quality of the paper. Part of this research was conducted while the first author was visiting the second author at the Universidad de Alcalá in early 2017, under the “GINER DE LOS RIOS” research program. Both authors wish to thank the Departamento de Física y Matemáticas de la Universidad de Alcalá for its support. The work of the three authors was partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under Grant MTM2015-65888-C4-2-P. The work of the first author was also supported by Conacyt’s Grant 287523.

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Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad de ColimaColimaMexico
  2. 2.Departamento de Física y MatemáticasUniversidad de AlcaláMadridSpain
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridMadridSpain
  4. 4.Instituto de Ciencias Matemáticas (ICMAT)MadridSpain

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