Delta hk
From timescalewiki
Let $\mathbb{T}$ be a time scale and let $t,s \in \mathbb{T}$. The $h_k$ monomials are defined by the recurrence $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_{n+1}(t,s)= \displaystyle\int_s^t h_{n}(\tau,s) \Delta \tau. \end{array} \right.$$
Graph of $h_2(t,0;\mathbb{Z})$.
Graph of $h_3(t,0;\mathbb{Z})$.
Graph of $h_4(t,0;\mathbb{Z})$.
Graph of $h_5(t,0;\mathbb{Z})$.
Contents
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
| $\mathbb{T}=$ | $h_k(t,s;\mathbb{T})=$ |
| $\mathbb{R}$ | $\dfrac{(t-s)^k}{k!}$ |
| $\mathbb{Z}$ | $\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$ |
| $h\mathbb{Z}$ | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
See also
$\Delta$-special functions on time scales | ||||||
 $\cos_p$ |
 $\cosh_p$ |
 $e_p$ |
 $g_k$ |
 $h_k$ |
 $\sin_p$ |
 $\sinh_p$ |
