37 research outputs found
CMV matrices in random matrix theory and integrable systems: a survey
We present a survey of recent results concerning a remarkable class of
unitary matrices, the CMV matrices. We are particularly interested in the role
they play in the theory of random matrices and integrable systems. Throughout
the paper we also emphasize the analogies and connections to Jacobi matrices.Comment: Based on a talk given at the Short Program on Random Matrices, Random
Processes and Integrable Systems, CRM, Universite de Montreal, 200
A "missing" family of classical orthogonal polynomials
We study a family of "classical" orthogonal polynomials which satisfy (apart
from a 3-term recurrence relation) an eigenvalue problem with a differential
operator of Dunkl-type. These polynomials can be obtained from the little
-Jacobi polynomials in the limit . We also show that these polynomials
provide a nontrivial realization of the Askey-Wilson algebra for .Comment: 20 page
Jordan algebras and orthogonal polynomials
We illustrate how Jordan algebras can provide a framework for the
interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi
polynomials are eigenfunctions of a first order operator of Dunkl type. We
consider an algebra that has this operator (up to constants) as one of its
three generators and whose defining relations are given in terms of
anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We
show how the structure and recurrence relations of the big -1 Jacobi
polynomials are obtained from the representations of this algebra. We also
present ladder operators for these polynomials and point out that the big -1
Jacobi polynomials satisfy the Hahn property with respect to a generalized
Dunkl operator.Comment: 11 pages, 30 reference