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.

Keywords:

weighted shifts, hyponormality, semi-cubical ,hyponormality.