We introduce a powerful algebraic method for proving that Lucas analogues are polynomials with nonnegative coefficients. In particular, we factor a Lucas polynomial as
$\{n\}=\prod_{d|n} P_d(s,t)$, 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. This is joint work with Jordan Tirrell based on an idea of Richard Stanley.