Dynamical sequences: closure properties and automatic identity proving

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 $φ$ 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.

💡 Research Summary

The paper introduces the notion of a dynamical sequence over an algebraically closed field K. A dynamical sequence is defined by a rational dynamical system (X, φ) together with a rational observation map f : X → 𝔸¹ and an initial point x₀∈X whose forward orbit avoids all indeterminacy loci of φ and f. The sequence is given by
  a(n) = f(φⁿ(x₀)), n ≥ 0.
This simple definition captures many classical integer sequences (factorials, binomial coefficients, Catalan numbers, etc.) as well as far more sophisticated objects such as Somos sequences, elliptic divisibility sequences (EDS), and the families of Cⁿ‑finite and Dⁿ‑finite sequences recently introduced by Jiménez‑Pastor, Nuspl, and Pillwein.

Section 2 – Closure properties.
The authors prove that the class of dynamical sequences is closed under nine fundamental operations:

1. Addition – by taking the product variety X×Y and the map (x,y)↦f(x)+g(y).
2. Multiplication – similarly using (x,y)↦f(x)·g(y).
3. Partial sums – by adjoining an auxiliary coordinate p that accumulates the values of f along the orbit; the map χ(x,p) = (φ(x), p+f(x)) yields the cumulative sum after n steps.
4. Partial products – analogous to (3) with χ(x,p) = (φ(x), p·f(x)).
5. Shifts – simply replace the initial point by φⁱ(x₀).
6. Modification of initial values – realized by a disjoint union of finitely many copies of X and a piecewise‑defined map that mimics the original dynamics after a finite “pre‑segment”.
7. Subsequences along arithmetic progressions – using the iterate φᵈ and a suitable offset.
8. Floor‑indexing – by constructing a d‑tuple space and a cyclic shift map µ that rotates the coordinates, thereby producing a sequence a(⌊n/d⌋).
9. Interlacing – by combining d sequences a₀,…,a_{d‑1} with indicator sequences c_i(n) (which are themselves linear‑recurrence sequences, hence dynamical) and then using sums and products.

These constructions show that dynamical sequences form a K‑algebra under pointwise operations. The authors note that closure under convolution remains open (Question 1).

Section 3 – Inclusion of known families.
A central technical tool is Proposition 3.2, which states that if a sequence f(n) satisfies a rational recurrence whose right‑hand side is a rational function of a finite collection of already‑known dynamical sequences together with a fixed number of previous values of f, then f itself is dynamical. Using this, the paper establishes:

• Cⁿ‑finite and Dⁿ‑finite sequences (for any n ≥ 1) are dynamical. The proof proceeds by induction on n, reducing a D^{m+1}‑finite recurrence to a rational combination of D^{m}‑finite sequences and applying Proposition 3.2.
• Somos sequences (bilinear recurrences of order k) become dynamical provided they have only finitely many zero terms. The recurrence is encoded as a rational map φ on A^{k} and the output coordinate f extracts the newest term.
• Elliptic divisibility sequences are treated analogously; the defining nonlinear recurrence can be written as a rational map once the finitely many zero entries are removed.
• Sequences of the form λ·d^{n} and λ·P(n) (with P∈ℚ