Delta cosine

Let $\mathbb{T}$ be a time scale and let $t_0 \in \mathbb{T}$ and let $\mu p^2 \colon \mathbb{T} \rightarrow \mathbb{R}$ 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 $i=\sqrt{-1}$.

Properties

Theorem

The following formula holds: $$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ where $\cos^{\Delta}_p$ denotes the delta derivative of the delta cosine function and $\sin_p$ denotes the delta sine function.

Proof

Compute $$\begin{array}{ll} \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\ &= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\sin_p(t,t_0), \end{array}$$ as was to be shown. █

Theorem

The following formula holds: $$\cos_p^2(t,t_0)+\sin_p^2(t,t_0)=e_{\mu p^2}(t,t_0),$$ where $\cos_p$ denotes the $\Delta$-$\cos_p$ function and $\sin_p$ denotes the $\Delta$-$\sin_p$ function.

Relation to other special functions

Theorem

The following formula holds: $$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ where $\sin_p$ denotes the $\Delta$-$\sin_p$ function and $\cos_p$ denotes the $\Delta$-$\cos_p$ function.

Proof

Compute $$\begin{array}{ll} \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) \\ &= \dfrac{ip}{2i} ( e_{ip}(t,t_0) + e_{-ip}(t,t_0) ) \\ &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\ &= p\cos_p(t,t_0), \end{array}$$ as was to be shown. █

Examples

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) = $

$\Delta$-special functions on time scales
$\cos_p$	$\cosh_p$	$e_p$	$g_k$	$h_k$	$\sin_p$	$\sinh_p$