Delta derivative of classical polynomial
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Theorem
Let $\mathbb{T}$ be a time scale, let $\alpha \in \mathbb{R}$, let $m \in \mathbb{N}$, and define $f \colon \mathbb{T} \rightarrow \mathbb{R}$ by $f(t)=(t-\alpha)^m$. Then $$f^{\Delta}(t)=\displaystyle\sum_{j=0}^{m-1} (\sigma(t)-\alpha)^j (t-\alpha)^{m-1-j},$$ where $\sigma$ denotes the forward jump.
Proof
References
- Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales (2001)... (previous): Theorem 1.24(i)
