CMV matrices and little and big -1 Jacobi polynomials

2011/08/17 by Maxim Derevyagin, Derevyagin, Maxim, Luc Vinet +3 · 3 citations
Computer Science · Mathematics · Physics and Astronomy · #33C45 #33C47 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Matrix Theory and Algorithms #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1108.3535

openalex publication_date 2011/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.

Cited by

• Mirror symmetric polynomials orthogonal on the unit circle
• Double affine Hecke algebra of rank 1 and orthogonal polynomials on the unit circle
• Eigenvalue equations for sieved polynomials or proving Askey right again

Related

• A CMV connection between orthogonal polynomials on the unit circle and the real line
• CMV: the unitary analogue of Jacobi matrices
• The CMV bispectrality of the Jacobi polynomials on the unit circle
• CMV matrices: Five years after
• Orthogonal polynomials related to some Jacobi-type pencils