# Nabla sinh

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

# Properties

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

Proof: proof goes here █

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

## $\nabla$-special functions on time scales

$\nabla$-$\widehat{\cos}_p$$\nabla-\widehat{\cosh}_p$$\nabla$-$\widehat{e}_p$$\nabla-h_k$$\nabla$-$g_k$$\nabla-\widehat{\sin}_p$$\nabla$-$\widehat{\sinh}_p$