Delta sine
From timescalewiki
Let $\mathbb{T}$ be a time scale and let $t_0 \in \mathbb{T}$ and let $p \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,t_0)=\dfrac{e_{ip}(t,t_0)-e_{-ip}(t,t_0)}{2i}$$
| $\mathbb{T}$ | |
| $\mathbb{R}$ | $\sin_p(t,s)= $ |
| $\mathbb{Z}$ | $\sin_p(t,s) = \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$ |
| $h\mathbb{Z}$ | $\sin_p(t,s) = $ |
| $\mathbb{Z}^2$ | $\sin_p(t,s) = $ |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\sin_p(t,s) = $ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | $\sin_p(t,s) =$ |
| $\mathbb{H}$ | $\sin_p(t,s) = $ |
