Difference between revisions of "Delta Taylor's formula"

Revision as of 17:04, 15 January 2023

Theorem

Let $\mathbb{T}$ be a time scale and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then $$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$ where $h_k$ denotes the $h_k$ polynomials.

Proof

References