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 █

Theorem

If $q$ is continuous and $\mu$-regressive then $$e_q(t,s)=\hat{e}_{\frac{q^{\rho}}{1+q^{\rho}\nu}}(t,s)=\hat{e}_{\ominus_{\nu}(-q^{\rho})}(t,s),$$ where $e_q$ denotes the $\Delta$-exponential and $\hat{e}_q$ denotes the $\nabla$-exponential.

Proof

References

Theorem

If $p$ is continuous and $\nu$-regressive then $$\hat{e}_p(t,s)=e_{\frac{p^{\sigma}}{1-p^{\sigma}\nu}}(t,s)=e_{\ominus(-p^{\sigma})}(t,s),$$ where $\hat{e}_p$ denotes the $\nabla$-exponential and $e_p$ denotes the $\Delta$-exponential.

Proof

References

Examples

Let $p$ be a $\nu$-regressive function.

Time Scale $\nabla$-exponential Functions
$\mathbb{T}=$	$\hat{e}_{p}(t,s)=$
$\mathbb{R}$
$\mathbb{Z}$	$\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=s}^{t-1} \dfrac{1}{1-p(k)} &; t \gt s \\ 1 &; t=s \\ \prod_{k=t}^{s-1} (1-p(k)) &; t \lt s \end{array} \right.$
$h\mathbb{Z}$	$\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \dfrac{1}{1-hp(hk)} &; t \gt s \\ 1 &; t=s \\ \prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} (1-hp(hk)) &; t \lt s \end{array} \right.$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

Let $\alpha$ be a regressive constant.

Time Scale $\nabla$-exponential Functions
$\mathbb{T}=$	$\hat{e}_{\alpha}(t,s)=$
$\mathbb{R}$	$\hat{e}_{\alpha}(t,s)=e^{\alpha(t-s)}$
$\mathbb{Z}$	$\hat{e}_{\alpha}(t,s)=\left( \dfrac{1}{1-\alpha} \right)^{t-s}$
$h\mathbb{Z}$	$\hat{e}_{\alpha}(t,s;h)=\left( \dfrac{1}{ 1-\alpha h} \right)^{\frac{t-s}{h}}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$	$\hat{e}_{\alpha}(t,s;q)=\displaystyle\prod_{\xi \in [s,t)} \dfrac{1}{1-(q-1)\alpha \xi}; t>s$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$