*Abstract:* If
we replace the integers in the
binomial coefficient expression
n!/k!(n-k)! with their corresponding
Fibonacci numbers, the result is
remarkably always an
integer. Fibonacci numbers generalize
to Lucas polynomials in variables s,t, and any quotient of products of integers has a Lucas analogue obtained by replacing each integer n with the corresponding Lucas polynomial n. There has been interest in deciding when such expressions, which are a priori only rational functions, are actually polynomials with nonnegative coefficients. In this talk, 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, 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. This is joint work with Bruce Sagan.