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