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Relative Asymptotics of Matrix Orthogonal Polynomials for Uvarov Perturbations: The Degenerate Case

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Abstract

Let \({\alpha}\) be a square matrix of measures, and \({\left\{P_n(x; \alpha)\right\}_{n\geq 0}}\) the associated sequence of orthonormal matrix polynomials satisfying the three-term recurrence relation \({x P_n(x; \alpha) = A_{n+1}(\alpha)P_{n+1}(x; \alpha) + B_n(\alpha) P_n(x; \alpha) + A_n^{\ast}(\alpha)P_{n-1}(x; \alpha),}\) \({n \geq 0.}\) Let \({{\rm d}\beta(u) {\overset{ {\rm def} }{=}} {\rm d}\alpha(u) + M\delta(u - c)}\), where \({M}\) is a positive definite matrix, \({\delta (u - c)}\) is the Dirac measure supported at \({c}\) that is located outside the support of \({{\rm d}\alpha}\). We study the outer relative asymptotics of the sequence \({\left\{P_n(x; \beta)\right\}_{n\geq 0}}\) with respect to the sequence \({\left\{P_n(x; \alpha)\right\}_{n\geq 0}}\) under quite general assumption on the coefficients of the three-term recurrence relation \({\left\{A_{n}(\alpha) \right\}_{n\geq 0}}\) and \({\left\{B_{n}(\alpha) \right\}_{n\geq 0}}\).

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Authors and Affiliations

  1. Département de Mathématique et Statistique, Faculté Polydisciplianire, Université Abdel Maleek Essadi, Tétouan, Morocco

    Hossain O. Yakhlef

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Spain

    Francisco Marcellán

  3. Instituto de Ciencias Matemáticas (ICMAT), Calle Nicolás Cabrera n 13-15 Campus de Cantoblanco, UAM, 28049, Madrid, Spain

    Francisco Marcellán

Authors
  1. Hossain O. Yakhlef
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  2. Francisco Marcellán
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Correspondence to Hossain O. Yakhlef.

Additional information

The work of the second author (FM) has been partially supported by Dirección general de Investigación Cientf́acuteica y Técnica, Ministerio de Economía y Competitividad of Spain, grant MTM2012-36732-C03-01.

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Yakhlef, H.O., Marcellán, F. Relative Asymptotics of Matrix Orthogonal Polynomials for Uvarov Perturbations: The Degenerate Case. Mediterr. J. Math. 13, 3135–3153 (2016). https://doi.org/10.1007/s00009-016-0676-x

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  • DOI: https://doi.org/10.1007/s00009-016-0676-x

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