Difference between revisions of "Table:Delta hk"
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(Created page with "{| class="wikitable" |+Time Scale $h_k$ Monomials |- |$\mathbb{T}=$ |$h_k(t,t_0)=$ |- | $\mathbb{R}$ |$\dfrac{(t-t_0)^k}{k!}$ |- | $\mathbb{Z}$...") |
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{| class="wikitable" | {| class="wikitable" | ||
| − | |+Time Scale $h_k$ Monomials | + | |+[[Table:Time scale hk monomials|Time Scale $h_k$ Monomials]] |
|- | |- | ||
|$\mathbb{T}=$ | |$\mathbb{T}=$ | ||
| − | |$h_k(t, | + | |$h_k(t,s;\mathbb{T})=$ |
|- | |- | ||
|[[Real_numbers | $\mathbb{R}$]] | |[[Real_numbers | $\mathbb{R}$]] | ||
| − | |$\dfrac{(t- | + | |$\dfrac{(t-s)^k}{k!}$ |
|- | |- | ||
|[[Integers | $\mathbb{Z}$]] | |[[Integers | $\mathbb{Z}$]] | ||
| − | |$\displaystyle{t- | + | |$\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$ |
|- | |- | ||
|[[Multiples_of_integers | $h\mathbb{Z}$]] | |[[Multiples_of_integers | $h\mathbb{Z}$]] | ||
| − | | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h- | + | | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |
|- | |- | ||
| [[Square_integers | $\mathbb{Z}^2$]] | | [[Square_integers | $\mathbb{Z}^2$]] | ||
| Line 18: | Line 18: | ||
|- | |- | ||
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]] | ||
| − | | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ | + | | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ |
|- | |- | ||
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]] | ||
Latest revision as of 01:31, 24 September 2016
| $\mathbb{T}=$ | $h_k(t,s;\mathbb{T})=$ |
| $\mathbb{R}$ | $\dfrac{(t-s)^k}{k!}$ |
| $\mathbb{Z}$ | $\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$ |
| $h\mathbb{Z}$ | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
