Delta sinh

Let $p$ and $-\mu p^2$ be regressive functions. Then the $\Delta$ hyperbolic sine function is defined by $$\sinh_p(t,s) = \dfrac{e_p(t,s)-e_{-p}(t,s)}{2i}.$$

Properties

Theorem

Let $p\in$ $C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then $$\sinh^{\Delta}_p = p\cosh_p,$$ where $\sinh_p$ denotes the $\Delta$-$\sinh_p$ function and $\cosh_p$ denotes the $\Delta$-$\cosh_p$.

Theorem

The following formula holds: $$\cosh^2_p - \sinh^2_p = e_{-\mu p^2},$$ where $\cosh_p$ denotes the $\Delta$-$\cosh_p$ function, $\sinh_p$ denotes the $\Delta$-$\sinh_p$ function, and $e_p$ denotes the $\Delta$-$e_p$ function.

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 $$y(t) = c_1 \cosh_{\gamma}(t,s) + c_2 \sinh_{\gamma}(t,s).$$

Relation to other functions

Theorem

Let $p\in C_{rd}$. If $-\mu p^2 \in \mathcal{R}$, then $$\cosh^{\Delta}_p = p\sinh_p,$$ where $\cosh_p$ denotes the $\Delta$-$\cosh_p$ function and $\sinh_p$ denotes the $\Delta$-$\sinh_p$ function.