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The Ultimate Signs of Second-Order Holonomic Sequences

Fugen Hagihara, Akitoshi Kawamura

TL;DR

The paper advances the understanding of second-order holonomic sequences by delivering a complete classification of possible ultimate signs, governed by the type (loxodromic, hyperbolic, elliptic) of the (P, Q) recurrence. It links the sign problem to continued fractions through canonical numerators/denominators and establishes monotonic convergence frameworks that explain the asymptotic sign behavior, while providing a partial algorithm that solves the Ultimate Sign Problem for most rational-input cases and identifying conditions for a total algorithm. The results yield concrete constraints on sign patterns (lengths 1,2,3,4,6,8,12 for rational P, Q) and describe a sign-partition of the initial-value space into convex regions, with constructive methods to realize or rule out certain patterns. By combining these theoretical insights with algorithmic reductions to the Minimality Problem and creative telescoping, the work makes progress toward decidability and practical sign-detection for holonomic sequences, and clarifies the roles of continued fractions in this context.

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.

The Ultimate Signs of Second-Order Holonomic Sequences

TL;DR

The paper advances the understanding of second-order holonomic sequences by delivering a complete classification of possible ultimate signs, governed by the type (loxodromic, hyperbolic, elliptic) of the (P, Q) recurrence. It links the sign problem to continued fractions through canonical numerators/denominators and establishes monotonic convergence frameworks that explain the asymptotic sign behavior, while providing a partial algorithm that solves the Ultimate Sign Problem for most rational-input cases and identifying conditions for a total algorithm. The results yield concrete constraints on sign patterns (lengths 1,2,3,4,6,8,12 for rational P, Q) and describe a sign-partition of the initial-value space into convex regions, with constructive methods to realize or rule out certain patterns. By combining these theoretical insights with algorithmic reductions to the Minimality Problem and creative telescoping, the work makes progress toward decidability and practical sign-detection for holonomic sequences, and clarifies the roles of continued fractions in this context.

Abstract

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

Paper Structure

This paper contains 14 sections, 20 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Ultimate signs
  4. Connection to continued fractions
  5. Connection to monotonic convergence of continued fractions
  6. Computing the ultimate sign
  7. Input set admitting a total algorithm
  8. Proof of the Main Results
  9. Proof of \ref{['thm:main']}
  10. Proof of \ref{['thm:procedure', 'thm:problem_reduction']}
  11. Proof of the lemmas
  12. Proof of the Other Results
  13. Proof of \ref{['thm:subcfrac_converge']}
  14. Proof of \ref{['thm:NOW21', 'prop:unstable']}

Key Result

Corollary 4

Let $P, Q \in \mathbb{R}(x)$ be rational functions without poles or zeros in $\mathbb{N}$, and suppose that $P$ has ultimate sign $(+)$. Define $S_{P, Q} \subseteq \{+, -, 0 \}^* \cup \{ \bot \}$ as above.

Figures (2)

  • Figure 1: left: The partition of the space of the initial values into $I_{P_{1/2}, Q}(s_j)$ for $s_j$ in \ref{['equation: ultimate sign example elliptic rational']}. right: The partition of the space of the initial values into $I_{P_{1/ \sqrt{2}}, Q}(s_j)$ for $s_j$ in \ref{['equation: 8 ultimate signs']}. In both pictures, the critical half line between $I_{P_t, Q}(s_j)$ and $I_{P_t, Q}(s_{j+1})$ belongs to $I_{P, Q}(s_{j+1})$ for each $j$, where $t= \frac{1}{2}, \frac{1}{\sqrt{2}}$, respectively.
  • Figure 2: The set of initial values $(f (0), f (1))$ of $\bigl( \frac{x+2}{x+1}, - \frac{x+3}{x+1} \bigr)$-holonomic sequences $f$ having each of the ultimate signs in \ref{['equation: ultimate sign example elliptic rational']}.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Corollary 4
  • Theorem 5
  • Example 6
  • Example 7
  • Corollary 8
  • Proposition 9
  • Theorem 10
  • ...and 18 more