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, resp., are classical. In a recent paper we observed that tridiagonal matrices and unitary Hessenberg matrices both belong to a wide 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.