Quantum q greater than 1
From timescalewiki
Let $q>1$. The set $\overline{q^{\mathbb{Z}}}=\{0, \ldots, q^{-2}, q^{-1}, 1, q, q^2, \ldots \}$ of quantum numbers is a time scale.
| Forward jump: | $\sigma(t)=qt$ | derivation |
| Forward graininess: | $\mu(t)=t(q-1)$ | derivation |
| Backward jump: | $\rho(t)=$ | derivation |
| Backward graininess: | $\nu(t)=$ | derivation |
| $\Delta$-derivative | $f^{\Delta}(t)= \left\{ \begin{array}{ll} \dfrac{f(qt)-f(t)}{t(q-1)} &; t\neq 0 \\ \displaystyle\lim_{h \rightarrow 0} \dfrac{f(h)-f(0)}{h} &; t=0 \end{array} \right.$ | derivation |
| $\nabla$-derivative | $f^{\nabla}(t)=$ | derivation |
| $\Delta$-integral | $\begin{array}{ll} \displaystyle\int_s^t f(\tau) \Delta \tau &= \displaystyle\sum_{k=\log_q(s)}^{\log_q(t)-1} q^{k} (q-1) f(q^k) \\ \end{array}$ | derivation |
| $\nabla$-integral | $\displaystyle\int_s^t f(\tau) \nabla \tau=$ | derivation |
| $h_k(t,s)$ | $h_k(t,s)=\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ | 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 |
| $e_p(t,s)$ | $e_p(t,s)=\displaystyle\prod_{k=\log_q(s)}^{\log_q(t)-1} 1 + p(q^k)q^{k}(q-1)$ | 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 | $\Gamma_{\overline{q^{\mathbb{Z}}}}(x,s)=$ | derivation |
| Euler-Cauchy logarithm | $L(t,s)=$ | derivation |
| Bohner logarithm | $L_p(t,s)=$ | derivation |
| Jackson logarithm | $\log_{\overline{q^{\mathbb{Z}}}} g(t)=$ | derivation |
| Mozyrska-Torres logarithm | $L_{\overline{q^{\mathbb{Z}}}}(t)=$ | derivation |
| Laplace transform | $\mathscr{L}_{\overline{q^{\mathbb{Z}}}}\{f\}(z;s)=$ | derivation |
| Hilger circle | derivation |
