Delta hk

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Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme: $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_{n+1}(t,s)= \displaystyle\int_s^t h_{n}(\tau,s) \Delta \tau. \end{array} \right.$$

Properties

Theorem

Let $\mathbb{T}$ be a time scale, let $t,s \in \mathbb{T}$, and let $k$ be a nonnegative integer. Then the following formula holds: $$h_k(t,s;\mathbb{T})=(-1)^kg_k(s,t;\mathbb{T}),$$ where $h_k$ denotes the delta hk and $g_k$ denotes the delta gk.

Proof

References

Examples

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

Delta gk

$\Delta$-special functions on time scales


$\cos_p$

$\cosh_p$

$e_p$

$g_k$

$h_k$

$\sin_p$

$\sinh_p$