Delta hk

From timescalewiki
Revision as of 18:32, 21 March 2015 by Tom (talk | contribs)
Jump to: navigation, search

Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme: $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_n(t,s)= \displaystyle\int_s^t h_{n-1}(\tau,s) \Delta \tau. \end{array} \right.$$

Time Scale $h_k$ Monomials
$\mathbb{T}=$ $h_k(t,s;\mathbb{T})=$
$\mathbb{R}$ $\dfrac{(t-s)^k}{k!}$
$\mathbb{Z}$ $\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$
$h\mathbb{Z}$ $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$ $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$