Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue probability density functions (p.d.f's) are β-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 β-ensemble, and we show how this joint density is related to known interrelations 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 β-ensemble.