Semi-cubically hyponormal weighted shifts with recursive type

In this paper, we discuss the semi-cubic hyponormality of recursively generated weighted shifts with weight α(x) : √x, (√a, √b, √c)^ to give a new bridge between cubically hyponormal and quadratically hyponormal weighted shifts. Using weight sequences with first two equal weights, we show that two notions of quadratic hyponormality and semi-cubic hyponormality are different one from another. Moreover, we characterize the semi-cubic hyponormality of weighted shifts.