Orthogonal matrix polynomials with orthogonal differences, Rodrigues� formulas and related subjects

Resumen

The content of this thesis is part of the areas of approximation theory and special functions, in particular, of the matrix orthogonality theory. It is composed by two parts. The first part is an introduction with the corresponding preliminaries, goals, summary of the results, discussion and conclusions. The second part is formed by two published papers: - A.J. Durán and V. Sánchez-Canales Orthogonal matrix polynomials whose differences are also orthogonal. J. Approx. Theory, (2014) 179, 112-127. - A. J. Durán and V. Sánchez-Canales Rodrigues� Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Difference Equations. Integral Transform. Spec. Funct. (2014) 25, 849-863. In these works we study four important properties of matrix orthogonal polynomials which are also eigenfunctions of a second order difference operator. We show what happens with four of the characterization properties of the classical discrete polynomials and the relations between all of them in the matrix case. We obtain new results that establish important differences between the scalar case and the matrix one: we prove that for orthogonal matrix polynomials the scalar characterizations for classical discrete families are no longer equivalent. More precisely, we prove that the equivalence between the orthogonality of the differences of orthogonal polynomials and the discrete Pearson equation for the associated weight matrix remains true for orthogonal matrix polynomials. Besides, under suitable Hermitian as assumptions, they also imply that the associated orthogonal polynomials are eigenfunctions of certain second order difference operator, but the converse is, in general, not true. We also study the question of the existence of Rodrigues� formulas for families of orthogonal matrix polynomials which are also eigenfunctions of a second order difference operator. We develop a method to find such formulas and, using it, we produce the first discrete Rodrigues� formulas in arbitrary size for two families of orthogonal matrix polynomials. Finally, we include two original and relevant examples of families of orthogonal matrix polynomials illustrating our results.