# Square integers

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The set $\mathbb{Z}^2 = \{0,1,4,9,16,\ldots\}$ of square integers is a time scale.

 Forward jump: $\sigma(t)=t+2\sqrt{t}+1$ derivation Forward graininess: $\mu(t)=2\sqrt{t}+1$ derivation Backward jump: $\rho(t)=$ derivation Backward graininess: $\nu(t)=$ derivation $\Delta$-derivative $f^{\Delta}(t)=\dfrac{f(t+2\sqrt{t}+1)-f(t)}{2\sqrt{t}-1}$ 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 $\mathbf{E}(t)=$ 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 $\Gamma_{\mathbb{Z}^2}(x,s)=$ derivation Euler-Cauchy logarithm $L(t,s)=$ derivation Bohner logarithm $L_p(t,s)=$ derivation Jackson logarithm $\log_{\mathbb{Z}^2} g(t)=$ derivation Mozyrska-Torres logarithm $L_{\mathbb{Z}^2}(t)=$ derivation Laplace transform $\mathscr{L}_{\mathbb{Z}^2}\{f\}(z;s)=$ derivation Hilger circle derivation