Lucas atoms

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≥2. An integer (e.g., a Catalan number) defined by an expression of the form ∏ini/∏jkj has a Lucas analogue obtained by replacing each factor with the correspo...

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Veröffentlicht in:Advances in mathematics (New York. 1965) 2020-11, Vol.374, p.107387, Article 107387
Hauptverfasser: Sagan, Bruce E., Tirrell, Jordan
Format: Artikel
Sprache:eng
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Zusammenfassung: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≥2. An integer (e.g., a Catalan number) defined by an expression of the form ∏ini/∏jkj 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}=∏d|nPd(s,t), where we call the polynomials Pd(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 Φd(q). Certain results about the Φd(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 Pd(s,t) at various specific values of the variables.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2020.107387