Square integers

The set $\mathbb{Z}^2 = \{0,1,4,9,16,\ldots\}$ of square integers is a time scale.

$\mathbb{T}=\mathbb{Z}^2$

Generic element $t\in \mathbb{T}$:	For some $n \in \mathbb{Z}, t=n^2$
Jump operator:	$\begin{array}{ll} \sigma(t) &= (\sqrt{t}+1)^2 \\ &= t + 2\sqrt{t} + 1 \end{array}$
Graininess operator:	$\begin{array}{ll} \mu(t) &= (t+2\sqrt{t}+1)-t \\ &= 2\sqrt{t}+1 \end{array}$
$\Delta$-derivative:	$f^{\Delta}(t)=\dfrac{f(t+2\sqrt{t}+1)-f(t)}{2\sqrt{t}-1}$
$\Delta$-integral:	$\displaystyle\int_s^t f(t) \Delta t = \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} (2k+1)f(k^2)$
Exponential function:	$\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1 + p(\tau) \mu(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} \mu(k^2) \dfrac{1}{\mu(k^2)} \log ( 1 + p(k^2)\mu(k^2)) \right) \\ &= \exp \left( \displaystyle\sum_{k=\sqrt{s}}^{\sqrt{t}-1} \log ( 1 + p(k^2)\mu(k^2)) \right) \\ &= \displaystyle\prod_{k=\sqrt{s}}^{\sqrt{t}-1} 1 + p(k^2)(2k+1) \end{array}$