Paper Info
Title
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) 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 (Φn W)(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.
Year
DOI
Venue
2005
10.1016/j.jat.2005.02.009
Journal of Approximation Theory
Keywords
Field
DocType
rodrigues formula,orthogonal polynomial,satisfiability,matrix polynomial,orthogonal polynomials
Rodrigues' formula,Laguerre polynomials,Classical orthogonal polynomials,Polynomial matrix,Mathematical analysis,Matrix (mathematics),Rodrigues' rotation formula,Legendre polynomials,Jacobi polynomials,Mathematics
Journal
Volume
Issue
ISSN
134
2
0021-9045
Citations 
PageRank 
References 
6
3.29
0
Authors
2
Name
Order
Citations
PageRank
Antonio J. Durán13613.73
F. Alberto Grünbaum2199.14