Zeros of Orthogonal Polynomials Generated by the Geronimus Perturbation of Measures

  • Amílcar Branquinho
  • Edmundo J. Huertas
  • Fernando R. Rafaeli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8579)

Abstract

This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure μ, i.e., (x − c)− 1(x) + (x − c), for some free parameter \(N\in{\rm{I\!R}}_{+}\) and shift c. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass N tends to infinity as well as we characterize the precise values of N such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure μ. When  μ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation c is located outside the support of μ.

Keywords

Orthogonal polynomials Canonical spectral transformations of measures Zeros Monotonicity Laguerre polynomials Asymptotic behavior Electrostatic interpretation Logarithmic potential 

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Amílcar Branquinho
    • 1
  • Edmundo J. Huertas
    • 1
  • Fernando R. Rafaeli
    • 2
  1. 1.CMUC and Departamento de Matemática (FCTUC)University of CoimbraCoimbraPortugal
  2. 2.Faculdade de MatemáticaUniversidade Federal de Uberlândia (UFU)UberlândiaBrazil

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