Publication: Zeros of Orthogonal Polynomials Generated by the Geronimus Perturbation of Measures
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2014
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Springer
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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)
−1dμ(x)+Nδ(x−c),
for some free parameter N ∈ IR+ 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.
Description
Proceedings of: 14th International Conference Computational Science and Its Applications (ICCSA 2014). Guimarães, Portugal, June 30 – July 3, 2014
Keywords
Orthogonal polynomials, Canonical spectral transformations of measures, Zeros, Monotonicity, Laguerre polynomials, Asymptotic behavior, Electrostatic interpretation, Logarithmic potential
Bibliographic citation
Computational Science and Its Applications - ICCSA 2014. Suiza: Springer, 2014, pp. 44-59