  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.
