Nabla cosh

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

Properties

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

Proof: █

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

References

Nabla dynamic equations