CMV matrices and little and big -1 Jacobi polynomials

Maxim Derevyagin, Luc Vinet, Alexei Zhedanov

Published 2011-08-17, updated 2011-08-22Version 2

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.