Nabla hk

Define the $\hat{h}_k$ monomials by $$\left\{ \begin{array}{ll} \hat{h}_0(t,s)=0 \\ \hat{h}_{n+1}(t,s)=\displaystyle\int_s^t \hat{h}_n(\tau,s) \nabla \tau. \end{array} \right.$$

• If $\mathbb{T}=\mathbb{R}$ we get $\hat{h}_k(t,s)=\dfrac{(t-s)^k}{k!}.$
• If $\mathbb{T}=\mathbb{Z}$ we get $\hat{h}_k(t,s)=\dfrac{(t-s)^{\overline{k}}}{k!}$.
• If $\mathbb{T}=\overline{q^{\mathbb{Z}}}, q>1$, then $\hat{h}_k(t,s)=\displaystyle\prod_{r=0}^{k-1} \dfrac{q^rt-s}{\sum_{j=0}^r q^j}$.

References

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