Difference between revisions of "Delta hk"

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Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme:
+
Let $\mathbb{T}$ be a [[time scale]] and let $t,s \in \mathbb{T}$. The $h_k$ Taylor monomials are defined by the recurrence
 
$$\left\{ \begin{array}{ll}
 
$$\left\{ \begin{array}{ll}
h_0(t,s)=1 \\
+
h_0(t,s;\mathbb{T})=1 \\
h_n(t,s)= \displaystyle\int_s^t h_{n-1}(\tau,s) \Delta \tau.
+
h_{k+1}(t,s;\mathbb{T})= \displaystyle\int_s^t h_{k}(\tau,s;\mathbb{T}) \Delta \tau.
 
\end{array} \right.$$
 
\end{array} \right.$$
  
{| class="wikitable"
+
<div align="center">
|+Time Scale $h_k$ Monomials
+
<gallery>
|-
+
File:Integerhk,k=2,s=0plot.png|Graph of $h_2(t,0;\mathbb{Z})$.
|$\mathbb{T}=$
+
File:Integerhk,k=3,s=0plot.png|Graph of $h_3(t,0;\mathbb{Z})$.
|$h_k(t,t_0)=$
+
File:Integerhk,k=4,s=0plot.png|Graph of $h_4(t,0;\mathbb{Z})$.
|-
+
File:Integerhk,k=5,s=0plot.png|Graph of $h_5(t,0;\mathbb{Z})$.
|[[Real_numbers | $\mathbb{R}$]]
+
</gallery>
|$\dfrac{(t-t_0)^k}{k!}$
+
</div>
|-
+
 
|[[Integers | $\mathbb{Z}$]]
+
=Properties=
|$\displaystyle{t-t_0 \choose k} = \dfrac{(t-t_0)!}{k! (t-t_0-k)!}$
+
[[Relationship between delta hk and delta gk]]<br />
|-
+
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
+
=Examples=
| $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-t_0)$
+
{{:Table:Delta hk}}
|-
+
 
| [[Square_integers | $\mathbb{Z}^2$]]
+
=See also=
|
+
[[Delta gk]]
|-
+
 
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
+
<center>{{:Delta special functions footer}}</center>
| $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^nt_0}{\sum_{i=0}^n q^i}$
+
 
|-
+
[[Category:specialfunction]]
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
+
[[Category:Definition]]
|
 
|-
 
|[[Harmonic_numbers | $\mathbb{H}$]]
 
|
 
|}
 

Latest revision as of 14:13, 28 January 2023

Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $h_k$ Taylor monomials are defined by the recurrence $$\left\{ \begin{array}{ll} h_0(t,s;\mathbb{T})=1 \\ h_{k+1}(t,s;\mathbb{T})= \displaystyle\int_s^t h_{k}(\tau,s;\mathbb{T}) \Delta \tau. \end{array} \right.$$

Properties

Relationship between delta hk and delta gk

Examples

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

See also

Delta gk

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$