Nabla exponential
The $\nabla$-exponential functions are examples of exponential functions on a time scale. 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 █
Examples
Let $\alpha$ be a regressive constant.
| $\mathbb{T}=$ | $\hat{e}_{\alpha}(t,s)=$ |
| $\mathbb{R}$ | $\hat{e}_{\alpha}(t,s)=e^{\alpha(t-s)}$ |
| $\mathbb{Z}$ | $\hat{e}_{\alpha}(t,s)=\left( \dfrac{1}{1-\alpha} \right)^{t-s}$ |
| $h\mathbb{Z}$ | $\hat{e}_{\alpha}(t,s;h)=\left( \dfrac{1}{ 1-\alpha h} \right)^{\frac{t-s}{h}}$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\hat{e}_{\alpha}(t,s;q)=\displaystyle\prod_{\xi \in [s,t)} \dfrac{1}{1-(q-1)\alpha \xi}; t>s$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
