Abstract
In this paper, we analyze a perturbation of a nontrivial probability measure dμ supported on an infinite subset on the real line, which consists on the addition of a time-dependent mass point. For the associated sequence of monic orthogonal polynomials, we study its dynamics with respect to the time parameter. In particular, we determine the time evolution of their zeros in the special case when the measure is semiclassical. We also study the dynamics of the Verblunsky coefficients, i.e., the recurrence relation coefficients of a polynomial sequence, orthogonal with respect to a nontrivial probability measure supported on the unit circle, induced from dμ through the Szegő transformation.
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In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium.
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Acknowledgements
The research of K. Castillo was supported by CNPq Program/Young Talent Attraction, Ministério da Ciência, Tecnologia e Inovação of Brazil, Project 370291/2013–1. The research of K. Castillo and F. Marcellán was supported by Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012–36732–C03–01. The research of L. Garza was supported by Conacyt (México) grant 156668 and PROMEP.
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Dedicated to Professor Hari M. Srivastava
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Castillo, K., Garza, L., Marcellán, F. (2014). On a Direct Uvarov-Chihara Problem and Some Extensions. In: Milovanović, G., Rassias, M. (eds) Analytic Number Theory, Approximation Theory, and Special Functions. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0258-3_25
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DOI: https://doi.org/10.1007/978-1-4939-0258-3_25
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