Let $0<q<1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{2}, q^{1}, 1, q^{-1}, q^{-2}, \ldots \}$ of quantum numbers is a time scale.
$\mathbb{T}=\overline{q^{\mathbb{Z}}}$
| Forward jump:
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$\sigma(t)=$
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derivation
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| Forward graininess:
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$\mu(t)=$
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derivation
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| Backward jump:
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$\rho(t)=$
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derivation
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| Backward graininess:
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$\nu(t)=$
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derivation
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| $\Delta$-derivative
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$f^{\Delta}(t)=$
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derivation
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| $\nabla$-derivative
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$f^{\nabla}(t)=$
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derivation
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| $\Delta$-integral
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$\displaystyle\int_s^t f(\tau) \Delta \tau=$
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derivation
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| $\nabla$-integral
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$\displaystyle\int_s^t f(\tau) \nabla \tau=$
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derivation
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| $h_k(t,s)$
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$h_k(t,s)=$
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derivation
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| $\hat{h}_k(t,s)$
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$\hat{h}_k(t,s)=$
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derivation
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| $g_k(t,s)$
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$g_k(t,s)=$
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derivation
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| $\hat{g}_k(t,s)$
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$\hat{g}_k(t,s)=$
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derivation
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| $e_p(t,s)$
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$e_p(t,s)=$
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derivation
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| $\hat{e}_p(t,s)$
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$\hat{e}_p(t,s)=$
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derivation
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| Gaussian bell
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$\mathbf{E}(t)=$
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derivation
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| $\mathrm{sin}_p(t,s)=$
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$\sin_p(t,s)=$
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derivation
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| $\mathrm{\sin}_1(t,s)$
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$\sin_1(t,s)=$
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derivation
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| $\widehat{\sin}_p(t,s)$
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$\widehat{\sin}_p(t,s)=$
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derivation
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| $\mathrm{\cos}_p(t,s)$
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$\cos_p(t,s)=$
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derivation
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| $\mathrm{\cos}_1(t,s)$
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$\cos_1(t,s)=$
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derivation
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| $\widehat{\cos}_p(t,s)$
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$\widehat{\cos}_p(t,s)=$
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derivation
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| $\sinh_p(t,s)$
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$\sinh_p(t,s)=$
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derivation
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| $\widehat{\sinh}_p(t,s)$
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$\widehat{\sinh}_p(t,s)=$
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derivation
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| $\cosh_p(t,s)$
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$\cosh_p(t,s)=$
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derivation
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| $\widehat{\cosh}_p(t,s)$
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$\widehat{\cosh}_p(t,s)=$
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derivation
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| Gamma function
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$\Gamma_{\overline{q^{\mathbb{Z}}}}(x,s)=$
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derivation
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| Euler-Cauchy logarithm
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$L(t,s)=$
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derivation
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| Bohner logarithm
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$L_p(t,s)=$
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derivation
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| Jackson logarithm
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$\log_{\overline{q^{\mathbb{Z}}}} g(t)=$
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derivation
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| Mozyrska-Torres logarithm
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$L_{\overline{q^{\mathbb{Z}}}}(t)=$
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derivation
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| Laplace transform
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$\mathscr{L}_{\overline{q^{\mathbb{Z}}}}\{f\}(z;s)=$
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derivation
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| Hilger circle
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derivation
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