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M. Derevyagin, L. Vinet and A. Zhedanov introduced in Constr. Approx. 36(2012) 513-535 a new connection between orthogonal polynomials on the unitcircle and the real line. It maps any real CMV matrix into a Jacobi onedepending on a real parameter $\lambda$. These authors prove that this mapyields a natural link between the Jacobi polynomials on the unit circle and thelittle and big $-1$ Jacobi polynomials on the real line. They also provideexplicit expressions for the measure and orthogonal polynomials associated withthe Jacobi matrix in terms of those related to the CMV matrix, but only for thevalue $\lambda=1$ which simplifies the connection -basic DVZ connection-.However, similar explicit expressions for an arbitrary value of $\lambda$ --(general) DVZ connection -- are missing. This is the main problem overcome inthis paper.  This work introduces a new approach to the DVZ connection which formulates itas a two-dimensional eigenproblem by using known properties of CMV matrices.This allows us to go further, providing explicit relations between the measuresand orthogonal polynomials for the general DVZ connection. It turns out thatthis connection maps a measure on the unit circle into a rational perturbationof an even measure supported on two symmetric intervals of the real line, whichreduce to a single interval for the basic DVZ connection, while theperturbation becomes a degree one polynomial. Some instances of the DVZconnection are shown to give new one-parameter families of orthogonalpolynomials on the real line.

A CMV connection between orthogonal polynomials on the unit circle and the real line