Difference between revisions of "Harmonic numbers"

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{| class="wikitable"
 
{| class="wikitable"
 
|+$\mathbb{T}=\mathbb{H}$
 
|+$\mathbb{T}=\mathbb{H}$
 +
|-
 +
|Generic element
 +
|If $t \in \mathbb{H}$, then for some positive integer $n$, $t=H_n=\displaystyle\sum_{k=0}^n \dfrac{1}{k}$.
 
|-
 
|-
 
|[[Forward jump]]:
 
|[[Forward jump]]:
|$\sigma(t)=$
+
|$\sigma(t)=\sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= t + \dfrac{1}{n+1}$
 
|[[Derivation of forward jump for T=Harmonic numbers|derivation]]
 
|[[Derivation of forward jump for T=Harmonic numbers|derivation]]
 
|-
 
|-
 
|[[Forward graininess]]:
 
|[[Forward graininess]]:
|$\mu(t)=$
+
|$\mu(t)=\mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= \dfrac{1}{n+1}$
 
|[[Derivation of forward graininess for T=Harmonic numbers|derivation]]
 
|[[Derivation of forward graininess for T=Harmonic numbers|derivation]]
 
|-
 
|-
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|-
 
|-
 
|[[Gaussian bell]]
 
|[[Gaussian bell]]
|$\mathbf{E}(t)=$
+
|$\mathbf{E}\left( H_n \dfrac{1}{k} \right)=\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$
 
|[[Derivation of Gaussian bell for T=Harmonic numbers|derivation]]
 
|[[Derivation of Gaussian bell for T=Harmonic numbers|derivation]]
 
|-
 
|-

Latest revision as of 00:38, 9 September 2015

The set $$\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\} = \left\{1,\frac{3}{2},\frac{11}{6},\frac{25}{12},\frac{137}{60},\frac{49}{20},\frac{ 363}{140},\frac{761}{280},\ldots \right\}$$ of harmonic numbers is a time scale.

$\mathbb{T}=\mathbb{H}$
Generic element If $t \in \mathbb{H}$, then for some positive integer $n$, $t=H_n=\displaystyle\sum_{k=0}^n \dfrac{1}{k}$.
Forward jump: $\sigma(t)=\sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= t + \dfrac{1}{n+1}$ derivation
Forward graininess: $\mu(t)=\mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= \dfrac{1}{n+1}$ derivation
Backward jump: $\rho(t)=$ derivation
Backward graininess: $\nu(t)=$ derivation
$\Delta$-derivative $f^{\Delta}(t)=$ derivation
$\nabla$-derivative $f^{\nabla}(t)=$ derivation
$\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=$ derivation
$\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=$ derivation
$h_k(t,s)$ $h_k(t,s)=$ 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
$e_p(t,s)$ $e_p(t,s)=$ derivation
$\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=$ derivation
Gaussian bell $\mathbf{E}\left( H_n \dfrac{1}{k} \right)=\displaystyle\prod_{k=1}^n \left( \dfrac{k}{k+1} \right)^{H_{k-1}}$ 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 $\Gamma_{\mathbb{H}}(x,s)=$ derivation
Euler-Cauchy logarithm $L(t,s)=$ derivation
Bohner logarithm $L_p(t,s)=$ derivation
Jackson logarithm $\log_{\mathbb{H}} g(t)=$ derivation
Mozyrska-Torres logarithm $L_{\mathbb{H}}(t)=$ derivation
Laplace transform $\mathscr{L}_{\mathbb{H}}\{f\}(z;s)=$ derivation
Hilger circle 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