Delta sine

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Let $\mathbb{T}$ be a time scale, let $s \in \mathbb{T}$, and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ be a regressive function. We define the trigonometric function $\sin_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by $$\sin_p(t,s;\mathbb{T})=\dfrac{e_{ip}(t,s;\mathbb{T})-e_{-ip}(t,s;\mathbb{T})}{2i}$$


Properties[edit]

Derivative of Delta sine
Sum of squares of delta cosine and delta sine
Derivative of delta cosine

Examples[edit]

Time Scale Sine Functions
$\mathbb{T}$ $\sin$$_p(t,s)= $
$\mathbb{R}$
$\mathbb{Z}$ $\dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$
$\Delta$-special functions on time scales

$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$