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Trajectories of Quadratic Differentials for Jacobi Polynomials with Complex Parameters

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Abstract

Motivated by the study of the asymptotic behavior of Jacobi polynomials \(\left( P_{n}^{(nA,nB)}\right) _{n}\) with \(A\in \mathbb {C}\) and \(B>0\), we establish the global structure of trajectories of the related rational quadratic differential on \(\mathbb {C}\). As a consequence, the asymptotic zero distribution \(\left( \text {limit of the root-counting measures of} \left( P_{n}^{(nA,nB)}\right) _{n}\right) \) is described. The support of this measure is formed by an open arc in the complex plane (critical trajectory of the aforementioned quadratic differential) that can be characterized by the symmetry property of its equilibrium measure in a certain external field.

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Notes

  1. Here, we understand by \(\gamma _{A,B}\) the open arc without its endpoints \(\zeta _\pm \).

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Acknowledgments

The first and second authors (AMF and PMG) were partially supported by MICINN of Spain and by the European Regional Development Fund (ERDF) under Grants MTM2011-28952-C02-01 and MTM2014-53963-P, by Junta de Andalucía (the research group FQM-229), and by Campus de Excelencia Internacional del Mar (CEIMAR) of the University of Almería. Additionally, AMF was supported by Junta de Andalucía through the Excellence Grant P11-FQM-7276. The part of this work was carried out during the visit of AMF to the Department of Mathematics of the Vanderbilt University. He acknowledges the hospitality of the host department, as well as a partial support of the Spanish Ministry of Education, Culture and Sports through the travel Grant PRX14/00037. We also wish to thank the anonymous referee for very useful remarks.

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Authors and Affiliations

  1. Department of Mathematics, University of Almería, Almería, Spain

    Andrei Martínez-Finkelshtein & Pedro Martínez-González

  2. Instituto Carlos I de Física Teórica y Computacional, Granada University, Granada, Spain

    Andrei Martínez-Finkelshtein

  3. ISSAT, University of Gabes, Gabés, Tunisia

    Faouzi Thabet

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  1. Andrei Martínez-Finkelshtein
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  2. Pedro Martínez-González
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  3. Faouzi Thabet
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Correspondence to Andrei Martínez-Finkelshtein.

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Communicated by Evguenii Rakhmanov.

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Martínez-Finkelshtein, A., Martínez-González, P. & Thabet, F. Trajectories of Quadratic Differentials for Jacobi Polynomials with Complex Parameters. Comput. Methods Funct. Theory 16, 347–364 (2016). https://doi.org/10.1007/s40315-015-0146-7

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  • DOI: https://doi.org/10.1007/s40315-015-0146-7

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