When can we decide that a P-finite sequence is positive?

TL;DR

This paper investigates two algorithms for proving the positivity of P-finite sequences, providing criteria for their guaranteed termination in certain cases, thus advancing the understanding of their applicability.

Contribution

It introduces a priori criteria that ensure the termination of two algorithms for positivity proofs of P-finite sequences of order up to three.

Findings

Algorithms succeed on many examples but do not always terminate.

A priori criteria guarantee termination for certain classes of P-finite recurrences.

Results improve understanding of algorithm applicability for positivity proofs.

Abstract

We consider two algorithms which can be used for proving positivity of sequences that are defined by a linear recurrence equation with polynomial coefficients (P-finite sequences). Both algorithms have in common that while they do succeed on a great many examples, there is no guarantee for them to terminate, and they do in fact not terminate for every input. For some restricted classes of P-finite recurrence equations of order up to three we provide a priori criteria that assert the termination of the algorithms.