Difference between revisions of "Table:Delta hk"

Latest revision as of 01:31, 24 September 2016

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

@@ Line 3: / Line 3: @@
 |-
 |$\mathbb{T}=$
-|$h_k(t,t_0)=$
+|$h_k(t,s;\mathbb{T})=$
 |-
 |[[Real_numbers | $\mathbb{R}$]]
-|$\dfrac{(t-t_0)^k}{k!}$
+|$\dfrac{(t-s)^k}{k!}$
 |-
 |[[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 | $h\mathbb{Z}$]]
-| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$
+| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$
 |-
 | [[Square_integers | $\mathbb{Z}^2$]]
@@ Line 18: / Line 18: @@
 |-
 |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
-| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$
+| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$
 |-
 |[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]