# 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 $$\cosh_p(t,s) = \dfrac{e_p(t,s)+e_{-p}(t,s)}{2}.$$
 $\mathbb{T}=$ $\cosh_1(t,0)=$ $\mathbb{R}$ $\cosh_1(t,0)=\cosh(t)$ $\mathbb{Z}$ $h\mathbb{Z}$ $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0)$ $\mathbb{Z}^2$ $\overline{q^{\mathbb{Z}}}, q > 1$ $\overline{q^{\mathbb{Z}}}, q < 1$ $\mathbb{H}$
 $\Delta$-special functions on time scales $\cos_p$ $\cosh_p$ $e_p$ $g_k$ $h_k$ $\sin_p$ $\sinh_p$