Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, I (original) (raw)

A survey on orthogonal matrix polynomials satisfying second order differential equations

Journal of Computational and Applied Mathematics, 2005

The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials.

Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples

Journal of Nonlinear Mathematical Physics, 2005

We describe a few families of orthogonal matrix polynomials of size N × N satisfying first order differential equations. This problem differs from the recent efforts reported for instance in [7] (Orthogonal matrix polynomials satisfying second order differential equations, Internat. Math. Research Notices, 2004 : 10 (2004), 461-484) and [15] (Matrix valued orthogonal polynomials of the Jacobi type, Indag. Math. 14 nrs. 3, 4 (2003), 353-366). While we restrict ourselves to considering only first order operators, we do not make any assumption as to their symmetry. For simplicity we restrict to the case N = 2. We draw a few lessons from these examples; many of them serve to illustrate the fundamental difference between the scalar and the matrix valued case.

An algebraic theory about semiclassical and classical matrix orthogonal polynomials

In this paper we introduce an algebraic theory of classical matrix orthogonal polynomials as a particular case of the semi-classical ones, defined by a distributional equation for the corresponding orthogonality functional. This leads to several properties that characterize the classical matrix families, among them, a structure relation and a second order differo-differential equation. In the particular case of Hermite type matrix polynomials we obtain all the parameters associated with the family and we prove that they satisfy, not only a differo-differential equation, but a second order differential one, as it can be seen in the scalar case.

On a new class of 2-orthogonal polynomials, II_The integral representations

arXiv (Cornell University), 2022

A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals (u 0 , u 1) was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties. Integral Transforms Spec Funct. 2021;32(2):134-153]. Six worthwhile special cases were pointed out there. Here we are precisely interesting with the integral representation problem for the functionals associated to these polynomials in each case. The focus will be on the matrix differential equation ΦU ′ + ΨU = 0, where Φ, Ψ are 2 × 2 polynomial matrices and U stands for the vector t (u 0 , u 1). We first establish the differential systems satisfied by the functionals u 0 and u 1 and then, depending on the case, we show that they admit integral representation via weight functions supported on the real line or on positive real line and defined in terms of various special functions. In order for certain integral representations to exist, addition of Dirac mass is necessary.

Differential properties of matrix orthogonal polynomials

2002

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(uA)=uB,D(u A) = u B,D(uA)=uB, where AAA and BBB 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.