2. Classifications of Recurrence Relations via Subclasses of (H, m)-quasiseparable Matrices

The results on characterization of orthogonal polynomials and Szegö polynomials via tridiagonal matrices and unitary Hessenberg matrices, respectively, are classical. In a recent paper we observed that tridiagonal matrices and unitary Hessenberg matrices both belong to a wider class of \((H,1)\)-quasiseparable matrices and derived a complete characterization of the latter class via polynomials satisfying certain EGO-type recurrence relations. We also established a characterization of polynomials satisfying three-term recurrence relations via \((H,1)\)-well-free matrices and of polynomials satisfying the Szegö-type two-term recurrence relations via \((H,1)\)-semiseparable matrices. In this paper we generalize all of these results from \(scalar\) (H,1) to the block (H, m) case. Specifically, we provide a complete characterization of \((H,\,m)\)-quasiseparable matrices via polynomials satisfying \(block\) EGO-type two-term recurrence relations. Further, \((H,\,m)\)-semiseparable matrices are completely characterized by the polynomials obeying \(block\) Szegö-type recurrence relations. Finally, we completely characterize polynomials satisfying m-term recurrence relations via a new class of matrices called \((H,\,m)\)-well-free matrices.