Dynamical sequences: closure properties and automatic identity proving

TL;DR

This paper introduces dynamical sequences over algebraically closed fields, demonstrating their closure properties, encompassing many classical sequences, and providing an algorithm for identity proving, with applications to combinatorial identities.

Contribution

It establishes the closure properties of dynamical sequences, shows they include many classical sequences, and develops an algorithm for proving sequence identities.

Findings

Dynamical sequences include elliptic divisibility, Somos, and $C^n$- and $D^n$-finite sequences.

The class of dynamical sequences is closed under various operations.

An algorithm is provided for proving identities between dynamical sequences.

Abstract

Given an algebraically closed field $K$, a dynamical sequence over $K$ is a $K$-valued sequence of the form $a(n):= f(\phi^n(x_0))$, where $\phi\colon X\to X$ 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 Jim\'enez-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.