Delta cosh

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Let $p \in C_{rd}$ and $-\mu p^2$ be a regressive function. Then the $\Delta$-hyperbolic cosine function is defined by $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$

Properties

Theorem: Let $p\in C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then $$\cosh^{\Delta}_p = p\sinh_p,$$ where $\sinh_p$ is the $\Delta$-hyperbolic sine function.

Proof:

Relation to other functions

Theorem: $\cosh^2_p - \sinh^2_p = e_{-\mu p^2}$

Proof:

Theorem: Let $\gamma$ be a nonzero regressive real number, then a general solution of the second order dynamic equation is $$y^{\Delta \Delta}-\gamma^2 y= 0$$ is given by $$y(t) = c_1 \cosh_{\gamma}(t,s) + c_2 \sinh_{\gamma}(t,s).$$

Proof:

Examples

Time Scale $\Delta$-$\cosh_1$ Functions
$\mathbb{T}=$ $\cosh_1(t,0)=$
$\mathbb{R}$ $\cosh_1(t,0)=\cosh(t)$
$\mathbb{Z}$
$h\mathbb{Z}$ $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$