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On Freud–Sobolev type orthogonal polynomials

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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 p, q 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.

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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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Authors and Affiliations

  1. Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Mexico

    Luis E. Garza

  2. Departamento de Física y Matemáticas, Universidad de Alcalá, Ctra. Madrid-Barcelona, Km. 33,600, Alcalá de Henares, 28871, Madrid, Spain

    Edmundo J. Huertas

  3. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida Universidad 30, Leganés, Madrid, Spain

    Francisco Marcellán

  4. Instituto de Ciencias Matemáticas (ICMAT), C/ Nicolás Cabrera, 13-15, Campus de Cantoblanco UAM, 28049, Madrid, Spain

    Francisco Marcellán

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  1. Luis E. Garza
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  2. Edmundo J. Huertas
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  3. Francisco Marcellán
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Correspondence to Luis E. Garza.

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Garza, L.E., Huertas, E.J. & Marcellán, F. On Freud–Sobolev type orthogonal polynomials. Afr. Mat. 30, 505–528 (2019). https://doi.org/10.1007/s13370-019-00663-6

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