Delta cosine

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 functions $\cos_p \colon \mathbb{T} \rightarrow \mathbb{R}$ $$\cos_p(t,t_0)=\dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2},$$ where $i=\sqrt{-1}$.

Time Scale Cosine Functions

$\mathbb{T}$
$\mathbb{R}$	$\cos_p(t,s)= $
$\mathbb{Z}$	$\cos_p(t,s) = $
$h\mathbb{Z}$	$\cos_p(t,s) = $
$\mathbb{Z}^2$	$\cos_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q > 1$	$\cos_p(t,s) = $
$\overline{q^{\mathbb{Z}}}, q < 1$	$\cos_p(t,s) =$
$\mathbb{H}$	$\cos_p(t,s) = $