A formula for $k$-hyponormality of backstep extensions of subnormal weighted shifts

Let ${\alpha }: {\alpha }_{0}, {\alpha }_{1}, \cdots$ be a weight sequence of positive real numbers and let $W_{{\alpha }}$ be a subnormal weighted shift with a weight sequence $\alpha$. Consider an extended weight sequence ${\alpha }(x) : x, {\alpha }_{0}, {\alpha }_{1}, \cdots$ with $0<x \le \alpha _{0}$and let $HE({\alpha ,k}):= \{ x >0: W_{\alpha (x)} \text{is} k \text{-hyponormal}\}$for $k \in {\mathbb{N}}\cup \{\infty \}$, where $\mathbb{N}$ is the set of natural numbers. We obtain a formula to find the interval $HE({\alpha ,k})\setminus HE({\alpha ,k+1})$, which provides several examples to distinguish the classes of $k$-hyponormal operators from one another.