Abstract:
In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with an hermitian functional. However, we give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case. A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Duran and Grunbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.

Abstract:
In this paper a general theory of semi-classical matrix orthogonal polynomials is developed. We define the semi-classical linear functionals by means of a distributional equation $D(u A) = u B,$ where $A$ and $B$ are matrix polynomials. Several characterizations for these semi-classical functionals are given in terms of the corresponding (left) matrix orthogonal polynomials sequence. They involve a quasi-orthogonality property for their derivatives, a structure relation and a second order differo-differential equation. Finally we illustrate the preceding results with some non-trivial examples.

Abstract:
Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl_{r+1} irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight. We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Pi\~neiro multiple orthogonal polynomial and others are given by Wronskian type determinants of Jacobi-Pi\~neiro polynomials. As a byproduct we obtain a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.

Abstract:
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix $r$ new rows and columns, so that the original Jacobi matrix is shifted downward. The $r$ new rows and columns contain $2r$ new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials and the $2r$ new parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.

Abstract:
In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced.

Abstract:
The relation between the spectral decomposition of a self-adjoint operator which is realizable as a higher order recurrence operator and matrix-valued orthogonal polynomials is investigated. A general construction of such operators from scalar-valued orthogonal polynomials is presented. Two examples of matrix-valued orthogonal polynomials with explicit orthogonality relations and three-term recurrence relation are presented, which both can be considered as $2\times 2$-matrix-valued analogues of subfamilies of Askey-Wilson polynomials.

Abstract:
The representations of the Schr\"odinger group in one space dimension are explicitly constructed in the basis of the harmonic oscillator states. These representations are seen to involve matrix orthogonal polynomials in a discrete variable that have Charlier and Meixner polynomials as building blocks. The underlying Lie-theoretic framework allows for a systematic derivation of the structural formulas (recurrence relations, difference equations, Rodrigues' formula etc.) that these matrix orthogonal polynomials satisfy.

Abstract:
It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. In this paper we extend this result and show that every system of polynomials satisfying some $(2N+1)$-term recurrence relation can be expressed in terms of orthonormal matrix polynomials for which the coefficients are $N\times N$ matrices. We apply this result to polynomials orthogonal with respect to a discrete Sobolev inner product and other inner products in the linear space of polynomials. As an application we give a short proof of Krein's characterization of orthogonal polynomials with a spectrum having a finite number of accumulation points.