Table:Delta hk

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Time Scale $h_k$ Monomials
$\mathbb{T}=$ $h_k(t,t_0)=$
$\mathbb{R}$ $\dfrac{(t-t_0)^k}{k!}$
$\mathbb{Z}$ $\displaystyle{t-t_0 \choose k} = \dfrac{(t-t_0)!}{k! (t-t_0-k)!}$
$h\mathbb{Z}$ $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$ $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$