Delta cosh
From timescalewiki
Let $p \in C_{rd}$ and $-\mu p^2$ be a regressive function. Then the $\Delta$-hyperbolic cosine function is defined by $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
Graph of $\cosh_{0.6}(t,0;\mathbb{Z})$.
Properties
Derivative of delta cosh
Delta cosh minus delta sinh
Delta hyperbolic trigonometric second order dynamic equation
Examples
| $\mathbb{T}=$ | $\cosh_1(t,0)=$ |
| $\mathbb{R}$ | $\cosh_1(t,0)=\cosh(t)$ |
| $\mathbb{Z}$ | |
| $h\mathbb{Z}$ | $\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) $ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
