  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.
