Difference between revisions of "Delta gk"

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(Created page with "==$g_k$ polynomials== $$g_0(t,s)=1$$ $$g_{n}(t,s) = \displaystyle\int_s^t g_{n-1}(\sigma(\tau),s) \Delta \tau$$ {| class="wikitable" |+Time Scale $g_k$ Monomials |- |$\mathbb...")
 
 
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==$g_k$ polynomials==
+
Let $\mathbb{T}$ be a [[time scale]] and let $t,s \in \mathbb{T}$. The $g_k$ monomials are defined by the recurrence
$$g_0(t,s)=1$$
+
$$\left\{ \begin{array}{ll}
$$g_{n}(t,s) = \displaystyle\int_s^t g_{n-1}(\sigma(\tau),s) \Delta \tau$$
+
g_0(t,s)=1 \\
 +
g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau.
 +
\end{array} \right.$$
  
{| class="wikitable"
+
<div align="center">
|+Time Scale $g_k$ Monomials
+
<gallery>
|-
+
File:Integergk,k=2,s=0plot.png|Graph of $g_2(t,0;\mathbb{Z})$.
|$\mathbb{T}=$
+
File:Integergk,k=3,s=0plot.png|Graph of $g_3(t,0;\mathbb{Z})$.
|$g_k(t,t_0)=$
+
File:Integergk,k=4,s=0plot.png|Graph of $g_4(t,0;\mathbb{Z})$.
|-
+
File:Integergk,k=5,s=0plot.png|Graph of $g_5(t,0;\mathbb{Z})$.
|[[Real_numbers | $\mathbb{R}$]]
+
</gallery>
|$g_k(t,t_0)=\dfrac{(t-t_0)^k}{k!}$
+
</div>
|-
+
 
|[[Integers | $\mathbb{Z}$]]
+
 
|$g_k(t,t_0)= $
+
 
|-
+
=Properties=
|[[Multiples_of_integers | $h\mathbb{Z}$]]
+
[[Zeros of delta gk]]<br />
| $g_k(t,t_0)=$
+
[[Relationship between delta hk and delta gk]]<br />
|-
+
 
| [[Square_integers | $\mathbb{Z}^2$]]
+
=Examples=
| $g_k(t,t_0)=$
+
{{:Table:Delta gk}}
|-
+
 
|[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
+
=See also=
| $g_k(t,t_0)=$
+
[[Delta hk]]
|-
+
 
|[[Quantum_q_less_than_1 | $\overline{q^{\mathbb{Z}}}, q &lt; 1$]]
+
<center>{{:Delta special functions footer}}</center>
| $g_k(t,t_0)=$
+
 
|-
+
[[Category:specialfunction]]
|[[Harmonic_numbers | $\mathbb{H}$]]
+
[[Category:Definition]]
|$g_k(t,t_0)=$
 
|}
 

Latest revision as of 14:13, 28 January 2023

Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $g_k$ monomials are defined by the recurrence $$\left\{ \begin{array}{ll} g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. \end{array} \right.$$


Properties

Zeros of delta gk
Relationship between delta hk and delta gk

Examples

Delta $g_k$ Monomials
$\mathbb{T}=$ $g_k(t,t_0)=$
$\mathbb{R}$ $g_k(t,t_0)=\dfrac{(t-t_0)^k}{k!}$
$\mathbb{Z}$ $g_k(t,t_0)= $
$h\mathbb{Z}$ $g_k(t,t_0)=$
$\mathbb{Z}^2$ $g_k(t,t_0)=$
$\overline{q^{\mathbb{Z}}}, q > 1$ $g_k(t,t_0)=$
$\overline{q^{\mathbb{Z}}}, q < 1$ $g_k(t,t_0)=$
$\mathbb{H}$ $g_k(t,t_0)=$

See also

Delta hk

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$