Delta hk

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Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $h_k$ 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[edit]

Relationship between delta hk and delta gk

Examples[edit]

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[edit]

Delta gk

$\Delta$-special functions on time scales

$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$