Revision as of 03:00, 28 March 2015

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

$\mathbb{T}=h\mathbb{Z}$

Generic element $t\in \mathbb{T}$:	For some $n \in \mathbb{Z}, t =hn$
Jump operator:	$\sigma(t)=t+h$
Graininess operator:	$\mu(t)=h$
$\Delta$-derivative:	$f^{\Delta}(t)=\dfrac{f(t+h)-f(t)}{h}$
$\nabla$-derivative:	$f^{\nabla}(t) = \dfrac{f(t)-f(t-h)}{h}$
$\Delta$-integral:	$\displaystyle\int_s^t f(\tau) \Delta \tau = \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} hf(hk)$
$h_k(t,s)$	$h_k(t,s) = \dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
$\Delta$-Exponential function:	$\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + hp(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=\frac{s}{h}}^{\frac{t}{h}-1} \log(1+hp(hk)) \right) \\ &= \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \left( 1+hp(hk) \right) \\ \end{array}$
$\nabla$-Exponential function:	$\hat{e}_p(t,s)=\left\{ \begin{array}{ll} \displaystyle\prod_{k=\frac{s}{h}}^{\frac{t}{h}-1} \dfrac{1}{1-hp(hk)} &; t \gt s \\ 1 &; t=s \\ \prod_{k=\frac{t}{h}}^{\frac{s}{h}-1} (1-hp(hk)) &; t \lt s \end{array} \right.$
Hilger circle:
Gamma function:	$\Gamma_{h\mathbb{Z}}(t;s)=h\displaystyle\sum_{k=0}^{\infty} \left( \displaystyle\prod_{j=s}^{k-1} \dfrac{j+x}{j+1} \right) \dfrac{1}{(1+h)^{k+1}}$

References

Cauchy Functions and Taylor's Formula for Time Scales T