Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f generalizing the circular beta-ensemble, and we show how this joint density is related to known inter-relations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi beta-ensemble