[[!redirects sequential antiderivatives]] #Contents# * table of contents {:toc} ## Definition ## Given a [[Z-module|$\mathbb{Z}$-module]] $M$ and a [[sequence]] $x:\mathbb{N} \to M$, the type of sequential antiderivatives of $x$ is the [[fiber]] of the [[sequential derivative]] at $x$: $$\mathrm{sequentialAntiderivatives}(x) \coloneqq \sum_{y:\mathbb{N} \to M} D(y) = x$$ A __sequential antiderivative__ is a term of the above type. ### In $\mathbb{Q}$-vector spaces ### If $M$ is a [[Q-vector space|$\mathbb{Q}$-vector space]], then every sequence in $M$ has an antiderivative operator for every term $m:M$, $$m:M \vdash D^{-1}_m:(\mathbb{N} \to M) \to (\mathbb{N} \to M)$$ defined as $$D^{-1}_m(x)(0) = m$$ $$D^{-1}_m(x)(i + i) \coloneqq \frac{x(i)}{i}$$ for $i:\mathbb{N}$. ## See also ## * [[sequence]] * [[sequential derivative]] * [[right shift operator]] category: not redirected to nlab yet