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]] | ||
|- | |- | ||
| Line 64: | Line 67: | ||
|- | |- | ||
|[[Gaussian bell]] | |[[Gaussian bell]] | ||
| − | |$\mathbf{E}( | + | |$\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.
| 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 |
