Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations |
| |
Authors: | Antonio J Durn F Alberto Grünbaum |
| |
Institution: | aDepartamento de Análisis Matemático, Universidad de Sevilla, Apdo (P.O. Box) 1160, 41080 Sevilla, Spain;bDepartment of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA |
| |
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) 461–484]). 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 (ΦnW)(n)W-1, where Φ 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. |
| |
Keywords: | Orthogonal matrix polynomials Orthogonal polynomials Rodrigues formula Pearson equation |
本文献已被 ScienceDirect 等数据库收录! |
|