On the integrality of some P-recursive sequences

TL;DR

This paper develops an algorithmic approach to determine the integrality and boundedness properties of P-recursive sequences, extending classical criteria and applying to sequences from the OEIS.

Contribution

It introduces a unified method and an algorithm for analyzing the global boundedness and integrality of solutions to second-order P-recursive sequences.

Findings

The algorithm can classify solutions as globally bounded or not.

It generalizes previous ad hoc methods for integrality analysis.

Successfully applied to multiple sequences from OEIS.

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).