Manuel Domínguez de la Iglesia

Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death Processes

We consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

\mathrm{SU}(N+1)

Differential equations for discrete Laguerre-Sobolev orthogonal polynomials

,

Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

\mathrm{U}(N)

Matrix-Valued Orthogonal Polynomials Related to SU(N+ 1), Their Algebras of Differential Operators, and the Corresponding Curves

); see [I. Pacharoni and J. A. Tirao, Constr. Approx., 25 (2007), pp. 177-192]. After an appropriate conjugation, we obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which we can explicitly compute its “n-step transition probability matrix” and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.

Non-Commutative Painlevé Equations and Hermite-Type Matrix Orthogonal Polynomials

Abstract It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form W ( t ) = t α e - t e At t B t B * e A * t , where A and B are certain (nilpotent and diagonal, respectively) N × N matrices. These weight matrices are the first examples illustrating this new phenomenon which are not reducible to scalar weights.

A note on the invariant distribution of a quasi-birth-and-death process

The aim of this paper is to study differential properties of orthogonal polynomials with respect to a discrete Laguerre-Sobolev bilinear form with mass point at zero. In particular we construct the orthogonal polynomials using certain Casorati determinants. Using this construction, we prove that they are eigenfunctions of a differential operator (which will be explicitly constructed). Moreover, the order of this differential operator is explicitly computed in terms of the matrix which defines the discrete Laguerre-Sobolev bilinear form.

Full length articles: Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We w ill show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar ca se. The combination of the ladder operators will lead to a family of second-order differentia l equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something a lso impossible in the scalar setting. This shows that the differential properties in the matrix ca se are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

On difference operators for symmetric Krall-Hahn polynomials

We give two examples of algebras of differential operators associated with families of matrix-valued orthogonal polynomials arising from representations of SU(N +1). The first gives a commutative algebra and the second a noncommutative one.

Second-Order Differential Operators Having Several Families of Orthogonal Matrix Polynomials as Eigenfunctions

We study double integral representations of Christoffel–Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its–Izergin–Korepin–Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painlevé IV differential equation for each family.