The Ultimate Signs of Second-Order Holonomic Sequences

TL;DR

This paper characterizes the long-term sign patterns of second-order holonomic sequences defined by linear recurrences with rational function coefficients, completing previous classifications and providing an algorithm for determining these patterns.

Contribution

It fully classifies all possible ultimate sign patterns of second-order holonomic sequences and extends prior work by covering more general cases with rational function coefficients.

Findings

Ultimate signs are limited to lengths 1, 2, 3, 4, 6, 8, or 12.

The sign pattern space is partitioned by initial conditions.

A partial algorithm can determine the ultimate sign in most cases.

Abstract

A real-valued sequence $f = \{ f(n) \}_{n \in \mathbb{N}}$ is said to be second-order holonomic if it satisfies a linear recurrence $f (n + 2) = P (n) f (n + 1) + Q (n) f (n)$ for all sufficiently large $n$, where $P, Q \in \mathbb{R}(x)$ are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of $f (n)$ follow for sufficiently large $n$. For each $P$, $Q$ we determine all the ultimate signs that $f$ can have, and show how they partition the space of initial values of $f$. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when $P$, $Q$ have rational coefficients, $f$ either has an ultimate sign of length $1$, $2$, $3$, $4$, $6$, $8$ or $12$, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of $f$ (or tells that there is none) in almost all cases.