On Semi-cubically Hyponormal Weighted Shifts with First Two Equal Weights

. It is known that a semi-cubically hyponormal weighted shift need not satisfy the flatness property, in which equality of two weights forces all or almost all weights to be equal. So it is a natural question to describe all semi-cubically hyponormal weighted shifts W α with first two weights equal. Let α : 1 , 1 , √ x, ( √ u, √ v, √ w ) ∧ be a backward 3-step extension of a recursively generated weight sequence with 1 < x < u < v < w and let W α be the associated weighted shift. In this paper we characterize completely the semi-cubical hyponormal W α satisfying the additional assumption of the positive determinant coefficient property, which result is parallel to results for quadratic hyponormality.