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Let σ be an l × l Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of matrix orthogonal polynomials associated with the Uvarov matrix transformation of σ defined by d σ u m (z) = d σ (z) + ∑ j = 1 m M j δ (z - ζ j) , where M j is an l × l positive definite matrix, ζ j ∈ C with ζ j ≠ ζ i and δ is the Dirac matrix measure. [ABSTRACT FROM AUTHOR]

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived. [ABSTRACT FROM AUTHOR]

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product where N ∈ ℝ, and a ∈ ℝ. We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler-Heine formula as well as a Plancherel-Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N. [ABSTRACT FROM AUTHOR]