# 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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Correspondence to Hossain O. Yakhlef.

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