Multiples of integers

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

HZtime scale.png

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


References[edit]

Cauchy Functions and Taylor's Formula for Time Scales T

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