Forward circle minus

Let $\mathbb{T}$ be a time scale and let $p,q \in \mathcal{R}(\mathbb{T},\mathbb{C})$ be (forward) regressive functions . We define the (forward) circle minus operation $\ominus_{\mu} \colon \mathbb{T} \rightarrow \mathbb{T}$ by $$\left( \ominus_{\mu} p \right)(t) = \dfrac{-p(t)}{1+p(t)\mu(t)}.$$ Since the set of forward regressive functions form a group $\left(\mathcal{R}(\mathbb{T},\mathbb{C}),\oplus_{\mu} \right)$ under circle plus with inverse operation $\ominus_{\mu}$, we define $$p \ominus_{\mu} q = p \oplus_{\mu} (\ominus_{\mu} q).$$

Properties

Forward regressive functions form a group

Theorem

The circle minus $\ominus_h$ is the inverse operation of the circle plus operation $\oplus_h$. Moreover, $$z \ominus_h w = z \oplus_h (\ominus_h w).$$