Computing with D-Algebraic Sequences

TL;DR

This paper formalizes the concept of D-algebraic sequences, explores their algebraic properties, and demonstrates how certain subsequences also satisfy algebraic difference equations, advancing understanding of their structural behavior.

Contribution

It provides a formal definition of D-algebraic sequences and analyzes their closure properties, including how subsequences indexed by arithmetic progressions inherit ADEs.

Findings

Subsequences indexed by arithmetic progressions satisfy ADEs of the same order.

D-algebraic sequences include holonomic and $C^2$-finite sequences with special difference-algebraic properties.

Algorithms for closure properties of D-algebraic sequences are proposed.

Abstract

A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations are denoted algebraic difference equations (ADEs). We propose a formal definition of D-algebraicity for sequences and investigate algorithms for their closure properties. We show that subsequences of D-algebraic sequences, indexed by arithmetic progressions, satisfy ADEs of the same orders as the original sequences. Additionally, we discuss the special difference-algebraic nature of holonomic and $C^2$-finite sequences.