Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the integrality of some P-recursive sequences

Anastasia Matveeva

Abstract

We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences due to Klazar and Luca, and propose a unified method for analysing global boundedness and algebraicity within a broader class of holonomic sequences. The central contribution is an algorithm that determines whether all, none, or a one-dimensional family of solutions to certain second-order recurrences are globally bounded. This approach generalizes earlier ad hoc methods and applies successfully to several well-known sequences from the On-Line Encyclopedia of Integer Sequences (OEIS).

On the integrality of some P-recursive sequences

Abstract

We investigate the arithmetic nature of P-recursive sequences through the lens of their D-finite generating functions. Building on classical tools from differential algebra, we revisit the integrality criterion for Motzkin-type sequences due to Klazar and Luca, and propose a unified method for analysing global boundedness and algebraicity within a broader class of holonomic sequences. The central contribution is an algorithm that determines whether all, none, or a one-dimensional family of solutions to certain second-order recurrences are globally bounded. This approach generalizes earlier ad hoc methods and applies successfully to several well-known sequences from the On-Line Encyclopedia of Integer Sequences (OEIS).

Paper Structure

This paper contains 16 sections, 16 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Integrality analysis of Motzkin-type sequences
  3. Approach of Klazar and Luca
  4. Discussion
  5. Arithmetic outlook
  6. Bridging global boundedness and algebraicity
  7. Hypergeometric case
  8. Second-order recurrences
  9. Deciding algebraicity of $I_1,I_2$.
  10. Transcendence in both components.
  11. Examples
  12. Integrality revisited
  13. Integrality criteria for small Apéry numbers
  14. Proof 1
  15. Proof 2
  16. ...and 1 more sections

Key Result

Theorem 1.3

(S80) A power series is D-finite if and only if its coefficient sequence is P-recursive.

Figures (4)

  • Figure 1:
  • Figure 2: Deciding whether $\int{x^n (1+a_1x+a_2x^2)^q dx}$, such that $a_1,a_2 \in \mathbb{Q}^*, n \in \mathbb{N}, q \in \mathbb{Q}$, is algebraic or transcendental. The constants $c, \tilde{c}, (A_1, B_1)$ come from \ref{['eq:4.4']}, \ref{['eq:4.7']}, \ref{['eq:4.8']}, respectively.
  • Figure 3: Deciding which of the cases \ref{['case:alg']}, \ref{['case:trans']}, \ref{['case:line']} holds for a recurrence relation of type \ref{['eq:4.1']} with $b_2 = \frac{2a_2b_1-a_1a_2b_0}{a_1}$. If the answer is \ref{['case:line']}, a pair $(s_0,s_1) \in \mathbb{Q}^2\setminus\{(0,0)\}$ such that $S(x)$ is algebraic is provided. The step "Decide algebraicity of $I_1,I_2$" is described in \ref{['fig:alg']}
  • Figure 4: Classification of the solution space of second-order linear recurrences with polynomial coefficients, according to integrality and global boundedness properties.

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 21 more