[[!redirects sequential derivatives]] #Contents# * table of contents {:toc} ## Idea ## A notion of derivative for [[sequences]] that behaves as the derivative does for the coefficients of power series. ## Definition ## Given a [[Z-module|$\mathbb{Z}$-module]] $M$ and a sequence $x:\mathbb{N} \to M$ of terms in $M$, the **sequential derivative** $$D:(\mathbb{N} \to M) \to (\mathbb{N} \to M)$$ is defined as $$D(x)(i) \coloneqq (i + 1) x(i)$$ for $i:\mathbb{N}$. ## See also ## * [[sequence]] * [[left shift operator]] * [[sequential antiderivative]] category: not redirected to nlab yet