# Difference between revisions of "Nth root of nonnegative integers"

Let $n \in \{1,2,\ldots\}$. The set $\sqrt[n]{\mathbb{N}_0}=\{0,1,\sqrt[n]{2}, \sqrt[n]{3},\ldots\}$ is a time scale.
 Forward jump: $\sigma(t)=\sqrt[n]{t^n+1}$ derivation Forward graininess: $\mu(t)=\sqrt[n]{t^n+1}-t$ derivation Backward jump: $\rho(t)=\sqrt[n]{t^n-1}$ derivation Backward graininess: $\nu(t)=t-\sqrt[n]{t^n-1}$ derivation $\Delta$-derivative $f^{\Delta}(t)=\dfrac{f(\sqrt[n]{t^n+1})-f(t)}{\sqrt[n]{t^n+1}-t}$ derivation $\nabla$-derivative $f^{\nabla}(t)=\dfrac{f(t)-f(\sqrt[n]{t^n-1})}{t-\sqrt{t^n-1}}$ derivation $\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\sum_{k=s^n}^{t^n-1} (\sqrt[n]{k+1}-\sqrt[n]{k}) f(\sqrt[n]{k})$ 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 $\mathbf{E}(t)=$ 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 $\Gamma_{\sqrt[n]{\mathbb{N}}}(x,s)=$ derivation Euler-Cauchy logarithm $L(t,s)=$ derivation Bohner logarithm $L_p(t,s)=$ derivation Jackson logarithm $\log_{\sqrt[n]{\mathbb{N}}} g(t)=$ derivation Mozyrska-Torres logarithm $L_{\sqrt[n]{\mathbb{N}}}(t)=$ derivation Laplace transform $\mathscr{L}_{\sqrt[n]{\mathbb{N}}}\{f\}(z;s)=$ derivation Hilger circle derivation