Semi-cubic hyponormality of weighted shifts with stampfli recursive tail

Let $\alpha :\sqrt{x_{m}},\cdots ,\sqrt{x_{1}},(\sqrt{u},\sqrt{v},\sqrt{w}% )^{\wedge}$ be a backward $m$-step extension of a recursive weight sequence and let $% W_{\alpha }$ be the weighted shift associated with $\alpha $. In this paper we characterize the semi-cubic hyponormality of $W_{\alpha }$ having the positive determinant coefficient property and discuss some related examples.