Integers

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The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.

$\mathbb{T}=\mathbb{Z}$
Forward jump: $\sigma(t)=t+1$ derivation
Forward graininess: $\mu(t)=1$ derivation
Backward jump: $\rho(t)=t-1$ derivation
Backward graininess: $\nu(t)=1$ derivation
$\Delta$-derivative $f^{\Delta}(t)=f(t+1)-f(t)$ derivation
$\nabla$-derivative $f^{\nabla}(t)=f(t)-f(t-1)$ derivation
$\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\int_s^t f(\tau) \Delta \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s}^{t-1} f(k) &; t \gt s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t}^{s-1} f(k) &; t \lt s \end{array} \right.$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=\displaystyle\int_s^t f(\tau) \nabla \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s+1}^t f(k) &; t>s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t+1}^s f(k) &; t\lt s \end{array} \right.$ derivation
$h_k(t,s)$ $h_k(t,s)=\dfrac{(t-s)^{\underline{k}}}{k!}$ derivation
$\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=\dfrac{(t-s)^{\overline{k}}}{k!}$ derivation
$g_k(t,s)$ $g_k(t,s)=$ derivation
$\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=$ derivation
$e_p(t,s)$ $e_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)} &; t < s \\ 1 &; t=s \\ \displaystyle\prod_{k=s}^{t-1} 1+p(k) &; t>s. \end{array} \right.$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=$ derivation
Gaussian bell $\mathbf{E}(t)=2^{\frac{-t(t-1)}{2}}$ derivation
$\mathrm{sin}_p(t,s)$ $\sin_p(t,s) = \left\{ \begin{array}{ll} \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} &; t>s \\ 0 &; t=s \\ \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i} &; t<s \end{array} \right.$ derivation
$\mathrm{\sin}_1(t,s)$ $\sin_1(t,s)=$ derivation
$\widehat{\sin}_p(t,s)$ $\widehat{\sin}_p(t,s)=$ derivation
$\mathrm{\cos}_p(t,s)$ $\cos_p(t,s)=\begin{array}{ll} \cos_p(t,t_0) &= \dfrac{e_{ip}(t,t_0)+e_{-ip}(t,t_0)}{2} \\ &= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) + \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2} \end{array}$ derivation
$\mathrm{\cos}_1(t,s)$ $\cos_1(t,s)=\begin{array}{ll} \cos_1(t,0) &= \dfrac{(1+i)^{t}+(1-i)^{t}}{2} \\ &= \dfrac{\displaystyle\sum_{k=0}^{t} {t \choose k} i^k + \displaystyle\sum_{k=0}^{t} (-1)^k {t \choose k} i^k}{2} \end{array}$ derivation
$\widehat{\cos}_p(t,s)$ $\widehat{\cos}_p(t,s)=$ derivation
$\sinh_p(t,s)$ $\sinh_p(t,s)=$ derivation
$\widehat{\sinh}_p(t,s)$ $\widehat{\sinh}_p(t,s)=$ derivation
$\cosh_p(t,s)$ $\cosh_p(t,s)=$ derivation
$\widehat{\cosh}_p(t,s)$ $\widehat{\cosh}_p(t,s)=$ derivation
Gamma function $\Gamma_{\mathbb{Z}}(x,s)=\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{2^{k+1}}$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\mathbb{Z}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{Z}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{Z}}\{f\}(z;s)=$ derivation
Hilger circle Hilgercircle,T=Z.png derivation

Examples of time scales

$\Huge\mathbb{R}$
Real numbers
$\Huge\mathbb{Z}$
Integers
$\Huge{h\mathbb{Z}}$
Multiples of integers
$\Huge\mathbb{Z}^2$
Square integers
$\Huge\mathbb{H}$
Harmonic numbers
$\Huge\mathbb{T}_{\mathrm{iso}}$
Isolated points
$\Huge\sqrt[n]{\mathbb{N}_0}$
nth root numbers
$\Huge\mathbb{P}_{a,b}$
Evenly spaced intervals
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q>1$
$\huge\overline{q^{\mathbb{Z}}}$
Quantum, $q<1$
$\overline{\left\{\dfrac{1}{n} \colon n \in \mathbb{Z}^+\right\}}$
Closure of unit fractions
$\Huge\mathcal{C}$
Cantor set