Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{A}^1$ are rational maps defined over $K$, and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $\varphi$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and we show that the class of dynamical sequences enjoys numerous closure properties and encompasses all elliptic divisibility sequences, all Somos sequences, and all $C^n$- and $D^n$-finite sequences for all $n\ge 1$, as defined by Jiménez-Pastor, Nuspl, and Pillwein. We also give an algorithm for proving that two dynamical sequences are identical and illustrate how to use this algorithm by showing how to prove several classical combinatorial identities via this method.

翻译：给定一个代数闭域$K$，$K$上的动力序列是指形如$a(n):= f(φ^n(x_0))$的$K$值序列，其中$φ\colon X\to X$与$f\colon X\to\mathbb{A}^1$是定义在$K$上的有理映射，且$x_0\in X$是一个其前向轨道避开$\varphi$和$f$的不确定点集的点。数论和代数组合学中的许多经典序列都属于此动力框架。我们证明了动力序列类具有丰富的闭包性质，并且包含了所有椭圆可除序列、所有Somos序列，以及所有由Jiménez-Pastor、Nuspl和Pillwein定义的、对所有$n\ge 1$成立的$C^n$-有限和$D^n$-有限序列。我们还给出了一种用于证明两个动力序列恒等的算法，并通过展示如何利用该方法证明若干经典组合恒等式来说明该算法的应用。