Delta hk
From timescalewiki
Define $h_n \colon \mathbb{T} \times \mathbb{T} \rightarrow \mathbb{R}$ by the scheme: $$\left\{ \begin{array}{ll} h_0(t,s)=1 \\ h_n(t,s)= \displaystyle\int_s^t h_{n-1}(\tau,s) \Delta \tau. \end{array} \right.$$
| $\mathbb{T}=$ | $h_k(t,s;\mathbb{T})=$ |
| $\mathbb{R}$ | $\dfrac{(t-s)^k}{k!}$ |
| $\mathbb{Z}$ | $\displaystyle{t-s \choose k} = \dfrac{(t-s)!}{k! (t-s-k)!}$ |
| $h\mathbb{Z}$ | $\dfrac{1}{k!} \displaystyle\prod_{\ell=0}^{k-1}(t-\ell h-s)$ |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | $\displaystyle\prod_{n=0}^{k-1} \dfrac{t-q^ns}{\sum_{i=0}^n q^i}$ |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
