Nabla cosh

$$\hat{\cosh}_p(t,s)=\dfrac{\hat{e}_p(t,s)+\hat{e}_{-p}(t,s)}{2}$$

Properties

Theorem: If $\alpha > 0$ with $\alpha^2\nu \in \mathcal{\nu}$, a regressive function, then $\hat{\cosh}_{\gamma}(\cdot,s)$ and $\hat{\sinh}_{\gamma}(\cdot,s)$ solve the $\nabla$ dynamic equation $$y^{\nabla \nabla}-\gamma^2 y=0.$$

Proof: proof goes here █

1. $\hat{\cosh}_p^{\nabla}(t,s)=p(t)\hat{\sinh}_p(t,s)$, where $\hat{\sinh}$ is the $\nabla$-$\sinh$ function.
2. $\hat{\cosh}^2_p(t,s)-\hat{\sinh}^2_p(t,s)=\hat{e}_{\nu p^2}(t,s)$
3. $\hat{\cosh}_p(t,s)-\hat{\sinh}_p(t,s)=\hat{e}_{-p}(t,s)$

References

Nabla dynamic equations