Difference between revisions of "Delta gk"

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==$g_k$ polynomials==
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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
$$g_0(t,s)=1$$
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$$\left\{ \begin{array}{ll}
$$g_{n}(t,s) = \displaystyle\int_s^t g_{n-1}(\sigma(\tau),s) \Delta \tau$$
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g_0(t,s)=1 \\
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g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau.
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\end{array} \right.$$
  
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<div align="center">
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<gallery>
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File:Integergk,k=2,s=0plot.png|Graph of $g_2(t,0;\mathbb{Z})$.
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File:Integergk,k=3,s=0plot.png|Graph of $g_3(t,0;\mathbb{Z})$.
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File:Integergk,k=4,s=0plot.png|Graph of $g_4(t,0;\mathbb{Z})$.
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File:Integergk,k=5,s=0plot.png|Graph of $g_5(t,0;\mathbb{Z})$.
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</gallery>
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</div>
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=Properties=
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[[Zeros of delta gk]]<br />
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[[Relationship between delta hk and delta gk]]<br />
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=Examples=
 
{{:Table:Delta gk}}
 
{{:Table:Delta gk}}
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=See also=
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[[Delta hk]]
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<center>{{:Delta special functions footer}}</center>
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[[Category:specialfunction]]
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[[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 $$\left\{ \begin{array}{ll} g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. \end{array} \right.$$


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


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$