Lucas atoms

TL;DR

This paper introduces algebraic methods to determine when Lucas polynomial expressions are actual polynomials, revealing their structure and connections to cyclotomic polynomials, and applies these to important combinatorial numbers.

Contribution

The paper develops a novel algebraic factorization approach for Lucas polynomials, enabling the proof that Lucas analogues of key combinatorial numbers are polynomials in s and t.

Findings

Lucas analogues of Fuss-Catalan and Fuss-Narayana numbers are polynomials in s,t.

Lucas atoms relate closely to cyclotomic polynomials, allowing transfer of properties.

The method generalizes classical theorems and provides explicit evaluation formulas.

Abstract

Given two variables $s$ and $t$, the associated sequence of Lucas polynomials is defined inductively by $\{0\}=0$, $\{1\}=1$, and $\{n\}=s\{n-1\}+t\{n-2\}$ for $n\ge2$. An integer (e.g., a Catalan number) defined by an expression of the form $\prod_i n_i/\prod_j k_j$ has a Lucas analogue obtained by replacing each factor with the corresponding Lucas polynomial. There has been interest in deciding when such expressions, which are a priori only rational functions, are actually polynomials in $s,t$. The approaches so far have been combinatorial. We introduce a powerful algebraic method for answering this question by factoring $\{n\}=\prod_{d|n} P_d(s,t)$, where we call the polynomials $P_d(s,t)$ Lucas atoms. This permits us to show that the Lucas analogues of the Fuss-Catalan and Fuss-Narayana numbers for all irreducible Coxeter groups are polynomials in $s,t$. Using gamma expansions, a technique which has recently become popular in combinatorics and geometry, one can show that the Lucas atoms have a close relationship with cyclotomic polynomials $\Phi_d(q)$. Certain results about the $\Phi_d(q)$ can then be lifted to Lucas atoms. In particular, one can prove analogues of theorems of Gauss and Lucas, deduce reduction formulas, and evaluate the $P_d(s,t)$ at various specific values of the variables.