Exponential functions

The classical exponential function $e^{x-s}$ is the unique solution to the initial value problem $$y'=y; y(s)=1.$$ The standard way to generalize this to time scales is called the $\Delta$-exponential function, which is the solution of $$y^{\Delta}=y;y(s)=1.$$ It generalizes the above equation in the sense that the classical derivative is replaced by the $\Delta$-derivative on some time scale. If instead of using the $\Delta$-derivative one uses the $\nabla$-derivative then the resulting exponential equation is $$y^{\nabla}=y;y(s)=1,$$ and we call its unique solution the $\nabla$-exponential function.

$\Delta$-exponential functions

Let $\mathbb{T}$ be a time scale. Define $\xi_h(z) := \dfrac{1}{h} \log(1+zh)$. Let $p \in \mathcal{R}(\mathbb{T},\mathbb{R})$ be a regressive function. The exponential function $e_p \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ is defined as

$$e_p(t,s) := \exp \left( \displaystyle\int_s^t \xi_{\mu(\tau)}(p(\tau))\Delta \tau \right)$$

for $s,t \in \mathbb{T}$. It turns out that $e_p$ is the unique solution to the dynamic initial value problem $$y^{\Delta} = py; y(s)=1.$$

Properties of Exponential Functions

For all $p,q \in \mathcal{R}$ and $t,s \in \mathbb{T}$,

• $e_p(t,r)e_p(r,s)=e_p(t,s)$ (semigroup property)
• $e_0(t,s)=1, e_p(t,t)=1$
• $e_p(\sigma(t),s)=(1+\mu(t)p(t))e_p(t,s)$
• $\dfrac{1}{e_p(t,s)}=e_{\ominus p}(s,t)$
• $e_p(t,s)e_q(t,s)=e_{p \oplus q}(t,s)$
• $\dfrac{e_p(t,s)}{e_q(t,s)} = e_{p \ominus q}(t,s)$
• $\left( \dfrac{1}{e_p(\cdot,s)} \right)^{\Delta} = -\dfrac{p(t)}{e_p^{\sigma}(\cdot,s)}$

Examples of $\Delta$-exponential Functions

• The Gaussian bell

Time Scale $\Delta$-exponential Functions

$\mathbb{T}=$	$e_p(t,s)=$
$\mathbb{R}$	$e_p(t,s)= \left\{ \begin{array}{ll} \exp \left( \displaystyle\int_s^t p(\tau) d \tau \right) &; t>s \\ 1 &; t=s \\ \exp \left( -\displaystyle\int_t^s p(\tau) d\tau \right) &; t<s class="wikitable"> 1$]]	$e_1(t,s)(t) = $
$\overline{q^{\mathbb{Z}}}, q < 1$	$e_1(t,s)(t) =$
$\mathbb{H}$	$e_1(t,s)(t) = $

$\nabla$-exponential Functions

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 █