卢卡斯原子
Lucas atoms
摘要：
给定两个变量$s$和$t$，卢卡斯多项式的相关序列由$n\ge2$的$\{0\}=0$、$\{1\}=1$和$\{n\}=s\{n-1\}+t\{n-2\}$归纳定义。整数（例如，由$\prod_i n_i/\prod_j k_j$形式的表达式定义的卡塔兰数）具有通过用相应的卢卡斯多项式替换每个因子而获得的卢卡斯模拟。有兴趣决定这种表达式，这是一个先验的唯一合理的功能，实际上是多项式在$s,t$。到目前为止，这些方法都是组合的。我们引入了一个强大的代数方法来回答这个问题，通过分解$\{n\}=\prod_{d|n} P_d(s,t)$，我们称之为多项式$P_d(s,t)$卢卡斯原子。这使我们能够表明，卢卡斯类似物的Fuss-Catalan和Fuss-Narayana数的所有不可约Coxeter群是多项式在$s,t$。使用伽玛展开，一种最近在组合数学和几何学中流行的技术，可以表明卢卡斯原子与分圆多项式$Φ_d(q)$有密切的关系。关于$Φ_d(q)$的某些结果可以被提升到卢卡斯原子。特别是，可以证明类似的定理高斯和卢卡斯，推导出减少公式，并评估$P_d(s,t)$在各种特定值的变量。
英文摘要：
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 $Φ_d(q)$. Certain results about the $Φ_d(q)$ 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.
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