On alternative definition of Lucas atoms and their $p$-adic valuations

TL;DR

This paper offers a new, more natural definition of Lucas atoms, fully characterizes their $p$-adic valuations for all primes, and shows that their sequence is not holonomic, advancing understanding of their algebraic properties.

Contribution

It introduces a more natural definition of Lucas atoms, characterizes their $p$-adic valuations for all primes, and proves the non-holonomicity of their sequence.

Findings

Lucas atoms can be defined more naturally with straightforward proofs.

Complete characterization of $p$-adic valuations for all primes.

Lucas atoms sequence is not holonomic.

Abstract

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.