Order bounds for $C^2$-finite sequences

TL;DR

This paper investigates $C^2$-finite sequences, which satisfy linear recurrences with $C$-finite coefficients, introduces new techniques for their closure properties, and derives novel order bounds using algebraic number theory.

Contribution

It presents new methods for closure properties of $C^2$-finite sequences and derives previously unknown order bounds using the exponent lattice of algebraic numbers.

Findings

Sequences satisfy closure properties similar to $C$-finite sequences.

New techniques enable derivation of order bounds.

Algorithm for computing bases of exponent lattices.

Abstract

A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.