Difference between revisions of "Delta hk"
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h_{n+1}(t,s)= \displaystyle\int_s^t h_{n}(\tau,s) \Delta \tau. | h_{n+1}(t,s)= \displaystyle\int_s^t h_{n}(\tau,s) \Delta \tau. | ||
\end{array} \right.$$ | \end{array} \right.$$ | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Integerhk,k=2,s=0plot.png|Graph of $h_2(t,0;\mathbb{Z})$. | ||
| + | File:Integerhk,k=3,s=0plot.png|Graph of $h_3(t,0;\mathbb{Z})$. | ||
| + | File:Integerhk,k=4,s=0plot.png|Graph of $h_4(t,0;\mathbb{Z})$. | ||
| + | File:Integerhk,k=5,s=0plot.png|Graph of $h_5(t,0;\mathbb{Z})$. | ||
| + | </gallery> | ||
| + | </div> | ||
{{:Table:Delta hk}} | {{:Table:Delta hk}} | ||
Revision as of 19:57, 1 June 2016
Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme: $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_{n+1}(t,s)= \displaystyle\int_s^t h_{n}(\tau,s) \Delta \tau. \end{array} \right.$$
Graph of $h_2(t,0;\mathbb{Z})$.
Graph of $h_3(t,0;\mathbb{Z})$.
Graph of $h_4(t,0;\mathbb{Z})$.
Graph of $h_5(t,0;\mathbb{Z})$.
| $\mathbb{T}=$ | $h_k(t,s;\mathbb{T})=$ |
| $\mathbb{R}$ | $\dfrac{(t-s)^k}{k!}$ |
| $\mathbb{Z}$ | $\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$ |
| $h\mathbb{Z}$ | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
