Difference between revisions of "Delta hk"

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(Created page with "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...")
 
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\end{array} \right.$$
 
\end{array} \right.$$
  
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{| class="wikitable"
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{{:Table:Time scale hk monomials
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|+Time Scale $h_k$ Monomials
 
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|$\mathbb{T}=$
 
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|$h_k(t,t_0)=$
 
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|[[Real_numbers | $\mathbb{R}$]]
 
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|$\dfrac{(t-t_0)^k}{k!}$
 
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|[[Integers | $\mathbb{Z}$]]
 
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|$\displaystyle{t-t_0 \choose k} = \dfrac{(t-t_0)!}{k! (t-t_0-k)!}$
 
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|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
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| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$
 
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| [[Square_integers | $\mathbb{Z}^2$]]
 
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|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q > 1$]]
 
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| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$
 
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|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q < 1$]]
 
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|[[Harmonic_numbers | $\mathbb{H}$]]
 
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Revision as of 18:32, 21 March 2015

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

{{:Table:Time scale hk monomials

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