Difference between revisions of "Delta hk"
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Revision as of 18:31, 21 March 2015
Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme: $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_n(t,s)= \displaystyle\int_s^t h_{n-1}(\tau,s) \Delta \tau. \end{array} \right.$$
| $\mathbb{T}=$ | $h_k(t,t_0)=$ |
| $\mathbb{R}$ | $\dfrac{(t-t_0)^k}{k!}$ |
| $\mathbb{Z}$ | $\displaystyle{t-t_0 \choose k} = \dfrac{(t-t_0)!}{k! (t-t_0-k)!}$ |
| $h\mathbb{Z}$ | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
