Difference between revisions of "Harmonic numbers"

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The set $\mathbb{H}=\left\{\displaystyle\sum_{k=1}^n \dfrac{1}{k} \colon n \in \mathbb{Z}^+ \right\}$ of harmonic numbers is a [[time scale]].
+
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]].
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|+$\mathbb{T}=\mathbb{H}$
 
|+$\mathbb{T}=\mathbb{H}$
 
|-
 
|-
−
|Generic element $t\in \mathbb{T}$:
+
|Generic element
−
|For some $n \in \mathbb{Z}^+, t =\displaystyle\sum_{k=0}^n \dfrac{1}{k}$
+
|If $t \in \mathbb{H}$, then for some positive integer $n$, $t=H_n=\displaystyle\sum_{k=0}^n \dfrac{1}{k}$.
 
|-
 
|-
−
|Jump operator:
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|[[Forward jump]]:
−
|$\begin{array}{ll}
+
|$\sigma(t)=\sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= t + \dfrac{1}{n+1}$
−
\sigma(t) &= \sigma \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\
+
|[[Derivation of forward jump for T=Harmonic numbers|derivation]]
−
&= \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \\
 
−
&= t + \dfrac{1}{n+1} \\
 
−
\end{array}$
 
 
|-
 
|-
−
|Graininess operator:
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|[[Forward graininess]]:
−
|$\begin{array}{ll}
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|$\mu(t)=\mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right)= \dfrac{1}{n+1}$
−
\mu(t) &= \mu \left( \displaystyle\sum_{k=1}^{n} \dfrac{1}{k} \right) \\
+
|[[Derivation of forward graininess for T=Harmonic numbers|derivation]]
−
&= t + \dfrac{1}{n+1} - t \\
 
−
&= \dfrac{1}{n+1} \\
 
−
\end{array}$
 
 
|-
 
|-
−
|[[Delta_derivative | $\Delta$-derivative:]]
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|[[Backward jump]]:
−
|$\begin{array}{ll}
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|$\rho(t)=$
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f^{\Delta}(t) &= f^{\Delta} \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right) \\
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|[[Derivation of backward jump for T=Harmonic numbers|derivation]]
−
&= \dfrac{f \left( \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \right) - f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right)}{\mu \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right)} \\
 
−
&= (n+1) \left[ f \left( \displaystyle\sum_{k=1}^{n+1} \dfrac{1}{k} \right) - f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right) \right]
 
−
\end{array}$
 
 
|-
 
|-
−
|[[Delta_integral | $\Delta$-integral:]]
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|[[Backward graininess]]:
−
| $\begin{array}{ll}
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|$\nu(t)=$
−
\displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\int_{\sum_{k=1}^m \frac{1}{k}}^{\sum_{k=1}^n \frac{1}{k}} f(\tau) \Delta \tau \\
+
|[[Derivation of backward graininess for T=Harmonic numbers|derivation]]
−
&= \displaystyle\sum_{k=m}^{n-1}\dfrac{1}{k+1} f \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k} \right)
+
|-
−
\end{array}$
+
|[[Delta derivative | $\Delta$-derivative]]
 +
|$f^{\Delta}(t)=$
 +
|[[Derivation of delta derivative for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla derivative | $\nabla$-derivative]]
 +
|$f^{\nabla}(t)=$
 +
|[[Derivation of nabla derivative for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta integral | $\Delta$-integral]]
 +
|$\displaystyle\int_s^t f(\tau) \Delta \tau=$
 +
|[[Derivation of delta integral for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla integral | $\nabla$-integral]]
 +
|$\displaystyle\int_s^t f(\tau) \nabla \tau=$
 +
|[[Derivation of nabla integral for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta hk|$h_k(t,s)$]]
 +
|$h_k(t,s)=$
 +
|[[Derivation of delta hk for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla hk|$\hat{h}_k(t,s)$]]
 +
|$\hat{h}_k(t,s)=$
 +
|[[Derivation of nabla hk for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta gk|$g_k(t,s)$]]
 +
|$g_k(t,s)=$
 +
|[[Derivation of delta gk for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla gk|$\hat{g}_k(t,s)$]]
 +
|$\hat{g}_k(t,s)=$
 +
|[[Derivation of nabla gk for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta exponential | $e_p(t,s)$]]
 +
|$e_p(t,s)=$
 +
|[[Derivation of delta exponential T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla exponential | $\hat{e}_p(t,s)$]]
 +
|$\hat{e}_p(t,s)=$
 +
|[[Derivation of nabla exponential T=Harmonic numbers|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 of Gaussian bell for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta sine | $\mathrm{sin}_p(t,s)=$]]
 +
|$\sin_p(t,s)=$
 +
|[[Derivation of delta sin sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|$\mathrm{\sin}_1(t,s)$
 +
|$\sin_1(t,s)=$
 +
|[[Derivation of delta sin sub 1 for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla sine|$\widehat{\sin}_p(t,s)$]]
 +
|$\widehat{\sin}_p(t,s)=$
 +
|[[Derivation of nabla sine sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta cosine|$\mathrm{\cos}_p(t,s)$]]
 +
|$\cos_p(t,s)=$
 +
|[[Derivation of delta cos sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|$\mathrm{\cos}_1(t,s)$
 +
|$\cos_1(t,s)=$
 +
|[[Derivation of delta cos sub 1 for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla cosine|$\widehat{\cos}_p(t,s)$]]
 +
|$\widehat{\cos}_p(t,s)=$
 +
|[[Derivation of nabla cos sub 1 for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta sinh|$\sinh_p(t,s)$]]
 +
|$\sinh_p(t,s)=$
 +
|[[Derivation of delta sinh sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla sinh|$\widehat{\sinh}_p(t,s)$]]
 +
|$\widehat{\sinh}_p(t,s)=$
 +
|[[Derivation of nabla sinh sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Delta cosh|$\cosh_p(t,s)$]]
 +
|$\cosh_p(t,s)=$
 +
|[[Derivation of delta cosh sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Nabla cosh|$\widehat{\cosh}_p(t,s)$]]
 +
|$\widehat{\cosh}_p(t,s)=$
 +
|[[Derivation of nabla cosh sub p for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Gamma function]]
 +
|$\Gamma_{\mathbb{H}}(x,s)=$
 +
|[[Derivation of gamma function for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Euler-Cauchy logarithm]]
 +
|$L(t,s)=$
 +
|[[Derivation of Euler-Cauchy logarithm for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Bohner logarithm]]
 +
|$L_p(t,s)=$
 +
|[[Derivation of the Bohner logarithm for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Jackson logarithm]]
 +
|$\log_{\mathbb{H}} g(t)=$
 +
|[[Derivation of the Jackson logarithm for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Mozyrska-Torres logarithm]]
 +
|$L_{\mathbb{H}}(t)=$
 +
|[[Derivation of the Mozyrska-Torres logarithm for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Laplace transform]]
 +
|$\mathscr{L}_{\mathbb{H}}\{f\}(z;s)=$
 +
|[[Derivation of Laplace transform for T=Harmonic numbers|derivation]]
 +
|-
 +
|[[Hilger circle]]
 +
|
 +
|[[Derivation of Hilger circle for T=Harmonic numbers|derivation]]
 
|-
 
|-
−
|[[Exponential_functions | Exponential function]]:
 
−
| $\begin{array}{ll}
 
−
e_p(t,s) &= e_p \left( \displaystyle\sum_{k=1}^n \dfrac{1}{k}, \displaystyle\sum_{k=1}^m \dfrac{1}{k} \right) \\
 
−
&= \exp \left( \displaystyle\int_{ \sum^m \frac{1}{k}}^{\sum^n \frac{1}{k}} \dfrac{1}{\mu(\tau)} \log(1 + \mu(\tau) p(\tau)) \Delta \tau \right) \\
 
−
&= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \mu \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\
 
−
&= \exp \left( \displaystyle\sum_{k=m}^{n-1} \log \left[1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \right] \right) \\
 
−
&= \displaystyle\prod_{k=m}^{n-1} 1 + \dfrac{1}{k+1} p \left( \displaystyle\sum_{j=1}^k \dfrac{1}{j} \right) \\
 
−
\end{array}$
 
 
|}
 
|}
 +
 +
<center>{{:Time scales footer}}</center>

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
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