Nabla exponential

Define the function $\hat{\xi}_{h} \colon \mathbb{C}_h \rightarrow \mathbb{Z}_h$ by $$\hat{\xi}_h(z) = \dfrac{1}{h} \log(1-zh).$$ Define the $\nabla$ exponential function for $s,t \in \mathbb{T}$ by $$\hat{e}_p(t,s) = \exp \left( \displaystyle\int_s^t \hat{\xi}_{\nu(\tau)}(p(\tau)) \nabla \tau \right).$$

Properties of $\nabla$-exponential functions

The function $\hat{e}_p(\cdot,s)$ is the unique solution of the initial value problem $$y^{\nabla} = py; y(s)=1.$$ For all $p,q \in \mathcal{R}_{\nu}$ and $t,s \in \mathbb{T}$,

• $\hat{e}_p(t,r)\hat{e}_p(r,s)=e_p(t,s)$ (semigroup property)
• $\hat{e}_0(t,s)=1, \hat{e}_p(t,t)=1$
• $\hat{e}_p(\rho(t),s)=(1-\nu(t)p(t))\hat{e}_p(t,s)$
• $\dfrac{1}{\hat{e}_p(t,s)}=\hat{e}_{\ominus_{\nu} p}(s,t)$
• $\hat{e}_p(t,s)\hat{e}_q(t,s)=\hat{e}_{p \oplus_{\nu} q}(t,s)$
• $\dfrac{\hat{e}_p(t,s)}{\hat{e}_q(t,s)} = \hat{e}_{p \ominus_{\nu} q}(t,s)$
• $\left( \dfrac{1}{\hat{e}_p(\cdot,s)} \right)^{\nabla} = -\dfrac{p(t)}{\hat{e}_p^{\rho}(\cdot,s)}$

Theorem: (Sign of the Nabla Exponential Function) Let $p \in \mathcal{R}_{\nu}$ and $s \in \mathbb{T}$.
i.) If $p \in \mathcal{R}_{\nu}^+$, then $\hat{e}_{p}(t,s) > 0$ for all $t \in \mathbb{T}$.
ii.) If $1-\nu(t)p(t) < 0$ for some $t \in \mathbb{T}_{\kappa}$, then $$\hat{e}(\rho(t),s)\hat{e}_{p}(t,s)<0.$$ iii.) If $1-\nu(t)p(t) < 0$ for all $t \in \mathbb{T}$, then $\hat{e}_p(t,s)$ changes sign at every point of $\mathbb{T}$.
iv.) The exponential function $\hat{e}_p(\cdot,s)$ is a real-valued function that is never equal to zero.

Proof: proof goes here █