Difference between revisions of "Table:Delta hk"

From timescalewiki
Jump to: navigation, search
 
(2 intermediate revisions by the same user not shown)
Line 3: Line 3:
 
|-
 
|-
 
|$\mathbb{T}=$
 
|$\mathbb{T}=$
|$h_k(t,t_0)=$
+
|$h_k(t,s;\mathbb{T})=$
 
|-
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|[[Real_numbers | $\mathbb{R}$]]
|$\dfrac{(t-t_0)^k}{k!}$
+
|$\dfrac{(t-s)^k}{k!}$
 
|-
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|[[Integers | $\mathbb{Z}$]]
|$\displaystyle{t-t_0 \choose k} = \dfrac{(t-t_0)!}{k! (t-t_0-k)!}$
+
|$\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$
 
|-
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$
+
| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
 
|-
 
|-
 
| [[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 | $\overline{q^{\mathbb{Z}}}, q > 1$]]
| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$
+
| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$
 
|-
 
|-
 
|[[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

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