Difference between revisions of "Delta gk"
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| − | + | 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_0(t,s)=1 \\ | |
| + | g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. | ||
| + | \end{array} \right.$$ | ||
| − | + | <div align="center"> | |
| − | + | <gallery> | |
| − | + | File:Integergk,k=2,s=0plot.png|Graph of $g_2(t,0;\mathbb{Z})$. | |
| − | |$\mathbb{ | + | File:Integergk,k=3,s=0plot.png|Graph of $g_3(t,0;\mathbb{Z})$. |
| − | |$ | + | File:Integergk,k=4,s=0plot.png|Graph of $g_4(t,0;\mathbb{Z})$. |
| − | + | File:Integergk,k=5,s=0plot.png|Graph of $g_5(t,0;\mathbb{Z})$. | |
| − | + | </gallery> | |
| − | |$ | + | </div> |
| − | + | ||
| − | + | ||
| − | |$ | + | |
| − | + | =Properties= | |
| − | + | [[Zeros of delta gk]]<br /> | |
| − | + | [[Relationship between delta hk and delta gk]]<br /> | |
| − | + | ||
| − | + | =Examples= | |
| − | + | {{:Table:Delta gk}} | |
| − | + | ||
| − | + | =See also= | |
| − | + | [[Delta hk]] | |
| − | + | ||
| − | + | <center>{{:Delta special functions footer}}</center> | |
| − | + | ||
| − | + | [[Category:specialfunction]] | |
| − | + | [[Category:Definition]] | |
| − | |||
| − | |||
Latest revision as of 14:13, 28 January 2023
Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $g_k$ monomials are defined by the recurrence $$\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.$$
Graph of $g_2(t,0;\mathbb{Z})$.
Graph of $g_3(t,0;\mathbb{Z})$.
Graph of $g_4(t,0;\mathbb{Z})$.
Graph of $g_5(t,0;\mathbb{Z})$.
Properties
Zeros of delta gk
Relationship between delta hk and delta gk
Examples
| $\mathbb{T}=$ | $g_k(t,t_0)=$ |
| $\mathbb{R}$ | $g_k(t,t_0)=\dfrac{(t-t_0)^k}{k!}$ |
| $\mathbb{Z}$ | $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
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
