Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations

Abstract

Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) ). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues' formulas of the type ( n W ) (n) W −1 , where is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues' formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues' formula and show that scalar type Rodrigues'formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it.

Dynamics and interpretation of some integrable systems via matrix orthogonal polynomials

“…In section 4, we give explicit expressions for the Weyl function, and we also obtain a representation of the vector functionals associated with the system studied in section 3. Moreover, we find the expression for the generalized Markov function (or equivalently, a measure of orthogonality) that governs this Toda type system and rediscovering, as an application, the one treated, independently by, other authors like Durán and Grünbaum in [17] and Miranian in [15].…”

“…From theorem 3.1 we know that if {a n , b n , c n , d n }, n ∈ N, is a solution of (1) it is equivalent to say that (17) is verified. Starting by (17), to obtain the relation (31) for R J it is sufficient to take derivatives in (5) and to substituteJ by its equivalent condition (17). Reciprocally, if (31) holds, using the fact that R J (z) = M F(z)M −1 and the paragraph (d) of theorem 3.1, we get {a n , b n , c n , d n }, n ∈ N, is a solution of (1).…”

“…theorem 2.1) that govern our high order Toda lattice are of type studied in [17], i.e. When U 1 do not depends on t, the Markov function that governs our Toda type system (1) with initial data associated with F 0 (z) given by (34) is of type F(z) = e z t F 0 (z) , which is the one studied in [15].…”

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

“…These have also been considered in [10] under the assumption that the weight is self-adjoint and positive semidefinite. A modified Rodrigues' formula, which seems to be better fit for the study of orthogonal matrix-valued polynomials was introduced and used in [11,13,12,9], yielding new families of polynomials.…”

“…Durán and Grünbaum conjectured that self-adjoint, positive semidefinite weights must reduce to scalar ones, in the sense that for some S we have W (x) = SΛ(x)S * with Λ(x) diagonal [10]. A proof of this conjecture appears in [3], where the result is deduced following the study of matrix orthogonal polynomials whose derivatives are also orthogonal.…”