Nabla exponential

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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 █

Relationship between delta exponential and nabla exponential
Relationship between nabla exponential and delta exponential


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

Time Scale $\nabla$-exponential Functions
$\mathbb{T}=$ $\hat{e}_{p}(t,s)=$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$

Let $\alpha$ be a regressive constant.

Time Scale $\nabla$-exponential Functions
$\mathbb{T}=$ $\hat{e}_{\alpha}(t,s)=$
$\mathbb{R}$ $\hat{e}_{\alpha}(t,s)=e^{\alpha(t-s)}$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$


Nabla dynamic equations on time scales

$\nabla$-special functions on time scales