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 $\cosh_p$ is the $\cosh_p$ 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: █