# Integers

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The set $\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$ of integers is a time scale.

 Forward jump: $\sigma(t)=t+1$ derivation Forward graininess: $\mu(t)=1$ derivation Backward jump: $\rho(t)=t-1$ derivation Backward graininess: $\nu(t)=1$ derivation $\Delta$-derivative $f^{\Delta}(t)=f(t+1)-f(t)$ derivation $\nabla$-derivative $f^{\nabla}(t)=f(t)-f(t-1)$ derivation $\Delta$-integral $\displaystyle\int_s^t f(\tau) \Delta \tau=\displaystyle\int_s^t f(\tau) \Delta \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s}^{t-1} f(k) &; t \gt s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t}^{s-1} f(k) &; t \lt s \end{array} \right.$ derivation $\nabla$-integral $\displaystyle\int_s^t f(\tau) \nabla \tau=\displaystyle\int_s^t f(\tau) \nabla \tau = \left\{ \begin{array}{ll} \displaystyle\sum_{k=s+1}^t f(k) &; t>s \\ 0 &; t=s \\ -\displaystyle\sum_{k=t+1}^s f(k) &; t\lt s \end{array} \right.$ derivation $h_k(t,s)$ $h_k(t,s)=\dfrac{(t-s)^{\underline{k}}}{k!}$ derivation $\hat{h}_k(t,s)$ $\hat{h}_k(t,s)=\dfrac{(t-s)^{\overline{k}}}{k!}$ derivation $g_k(t,s)$ $g_k(t,s)=$ derivation $\hat{g}_k(t,s)$ $\hat{g}_k(t,s)=$ derivation $e_p(t,s)$ $e_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=t}^{s-1} \dfrac{1}{1+p(k)} &; t < s \\ 1 &; t=s \\ \displaystyle\prod_{k=s}^{t-1} 1+p(k) &; t>s. \end{array} \right.$ derivation $\hat{e}_p(t,s)$ $\hat{e}_p(t,s)=$ derivation Gaussian bell $\mathbf{E}(t)=2^{\frac{-t(t-1)}{2}}$ derivation $\mathrm{sin}_p(t,s)$ \$\sin_p(t,s) = \left\{ \begin{array}{ll} \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i} &; t>s \\ 0 &; t=s \\ \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i} &; t