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 $$\left\{ \begin{array}{ll} g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau. \end{array} \right.$$


Properties[edit]

Zeros of delta gk
Relationship between delta hk and delta gk

Examples[edit]

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[edit]

Delta hk

$\Delta$-special functions on time scales

$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$