Matrix orthogonal polynomials whose derivatives are also orthogonal

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.Comment: 49 page

The Algebra of Differential Operators for a Gegenbauer Weight Matrix

In this paper we study in detail algebraic properties of the algebra $\mathcal D(W)$ of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed $\mathcal D(W)$ is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra $\mathcal D(W)$ is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra $\mathcal D(W)$ and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra $\mathcal D(W)$ as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction

Matrix orthogonal polynomials satisfying second-order differential equations: Coping without help from group representation theory

AbstractThe method developed in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484] led us to consider polynomials that are orthogonal with respect to weight matrices W(t) of the form e-t2T(t)T*(t), tαe-tT(t)T*(t) and tα(1-t)βT(t)T*(t), with T satisfying T′=(2Bt+A)T, T(0)=I, T′=(A+B/t)T, T(1)=I and T′(t)=(A/t+B/(1-t))T, T(1/2)=I, respectively. Here A and B are in general two non-commuting matrices. To proceed further and find situations where these polynomials satisfied second-order differential equations, we needed to impose commutativity assumptions on the pair of matrices A,B. In fact, we only dealt with the case when one of the matrices vanishes.The only exception to this arose as a gift from group representation theory: one automatically gets a situation where A and B do not commute, see Grünbaum et al. [Matrix valued orthogonal polynomials of the Jacobi type: the role of group representation theory, Ann. Inst. Fourier Grenoble 55 (6) (2005) 2051–2068]. This corresponds to the last of the three cases mentioned above.The purpose of this paper is to consider the other two situations and since now we do not get any assistance from representation theory we make a direct attack on certain differential equations in Duran and Grünbaum [Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Res. Notices 10 (2004) 461–484]. By solving these equations we get the appropriate weight matrices W(t), where the matrices A,B give rise to a solvable Lie algebra

The Analytic Theory of Matrix Orthogonal Polynomials

We give a survey of the analytic theory of matrix orthogonal polynomials.Comment: 85 page

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to 2×2 Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2×2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix

Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.Comment: 24 page

Riemann--Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement

In this paper matrix orthogonal polynomials in the real line are described in terms of a Riemann--Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by recursion coefficients to quartic Freud matrix orthogonal polynomials or not.Comment: 22 page