Abstract:

In this thesis we present a series of result that are framed in the theory of Matrix Orthogonal Polynomials, a branch of the very celebrated subject of Orthogonal Polynomials. In particular we study families of matrix valued orthogonal polynomials s...
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In this thesis we present a series of result that are framed in the theory of Matrix Orthogonal Polynomials, a branch of the very celebrated subject of Orthogonal Polynomials. In particular we study families of matrix valued orthogonal polynomials satisfying differential, difference or qdifference equations. The search for examples of matrix polynomials which are also eigenfunctions of certain matrix operators is a rather dificult issue. This dificulty is due to several reasons. The most important of these reasons is the increase of the difficulties in the computations related with the noncommutativity of the matrix product, and the existence of singular matrices. However, having a wide set of examples of matrix orthogonal polynomials has shown to be decisive in the study and discovery of new phenomena happening in the matrix orthogonality. This has been the case with the examples of matrix orthogonal polynomials satisfying differential equations. An example of the increasing knowledge about these families of orthogonal polynomials is shown in the last chapter of this memory. There, lot of tools developed in the last decades are used to build and study in deep an interesting family of matrix orthogonal polynomials satisfying second order differential equations. Moreover, these families are shown to satisfy first order differential equations as well. In the case of matrix orthogonal polynomials satisfying difference equations, very little was known. Apart from some examples in size 2x2 (and some others reducible to the scalar case) there were no examples of such matrix orthogonal polynomials. With this thesis this lack of examples starts to be solved. Moreover, we introduce a method to construct examples of matrix orthogonal polynomials satisfying second order difference equations, and by making use of it we give a variety of examples. Having such a method is of the main importance, because we skip the complexity in the computations that made the search of examples so difficult, and now dealing with matrix orthogonal polynomials and matrix difference equations becomes much easier to handle. The method profits of the factorization of a weight matrix and the symmetry equations for a discrete weight matrix and a difference operator. These symmetry equations are the starting point to develop the method. By making use of the examples obtained by this method, we explore new features and properties satisfied by this objects. For the case of matrix orthogonal polynomials satisfying qdifference equations, even less was known. In this thesis we establish the symmetry equations for the qdifference case, and we adapt the method developed for the difference case to obtain the first nontrivial examples of matrix orthogonal polynomials satisfying second order qdifference operator. That emphasizes the power of the method to build examples for the difference case. With our method we construct an example of matrix orthogonal polynomials satisfying qdifference equations, but this method can easily be used to get a wider class of examples and to explore their properties. With the content of this thesis we get a more complete view of the theory of matrix orthog onal polynomials, and many questions can now be tackled, such as those concerning limiting relations among matrix orthogonal polynomials satisfying second order difference equations (or qdifference) and matrix orthogonal polynomials satisfying second order differential equa tions.
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