Lucas atoms
Bruce E. Sagan (Michigan State University), Jordan Tirrell, (Washington College)

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 and , the associated sequence of Lucas polynomials is defined inductively by , , and for . An integer (e.g., a Catalan number) defined by an expression of the form 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 . The approaches so far have been combinatorial. We introduce a powerful algebraic method for answering this question by factoring , where we call the polynomials 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 . Using gamma expansions, a…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Theories and Applications · Advanced Mathematical Identities
