Variations on a circular Hessenberg pair

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a Hessenberg fashion. A Hessenberg system $\Phi$ an `oriented' version of a Hessenberg pair. It is known that $\Phi$ is determined up to isomorphism by its parameter array; this consists of the eigenvalue sequence of $\Phi$, the dual eigenvalue sequence of $\Phi$, and a sequence of nonzero scalars $\{\phi_i\}_{i=1}^d$ called the split sequence of $\Phi$. We are interested in some types of Hessenberg matrices, said to be circular, quasi-circular, tridiagonal, and irreducible tridiagonal. We are interested in the families of Hessenberg systems for which the associated Hessenberg matrices have one of the above types. A Hessenberg system of irreducible tridiagonal type is often called a Leonard system. In this case the associated Hessenberg pair
satisfies two relations, called the tridiagonal relations. We are interested in the family of Hessenberg systems for which the associated Hessenberg pair satisfies the tridiagonal relations. We are also interested in the family of Hessenberg systems for which the eigenvalue sequence and dual eigenvalue sequence
satisfy a linear three-term recurrence. In the present paper we have two main goals. First, we show how the above families of Hessenberg systems are related to each other. Second, we describe each family in terms of the parameter array.