# Harmonic numbers

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