Lucas atoms

Positivity of Narayana polynomials and Eulerian polynomials

“…Notably, γ-positivity of a polynomial implies that its coefficients are symmetric and unimodal, and the coefficients of γ-positive polynomials often have nice geometric and combinatorial interpretations, see [1,39] for details. If the γ-vector of f (x) alternates in sign, then we say that f (x) is alternatingly γ-positive (see [11,31,45] for instance). For example, (1 + x 2 ) n is alternatingly γ-positive, since…”

“…where the alternating γ-coefficients n k 2 k count k-simplices in the n-cube (see [48, A013609]). There has been considerable recent interest in the study alternatingly γ-positive polynomials, see [11,31,34,45] for instance. For example, Lin etal.…”

“…The reader is referred to (8) for the alternating γ-expansion of {n + 1}. Sagan and Tirrell [45] introduced a sequence of polynomials P n (s, t) by using the factorization of {n}:…”

“…The polynomials P n (s, t) are called Lucas atoms. Motivated by (4), Sagan and Tirrell [45] first established a connection between cyclotomic polynomials and Lucas atoms, and then proved the alternating γ-positivity of cyclotomic polynomials. They also wrote in their paper [45, p. 24]:"it might also be interesting to look at gamma expansions where the coefficients alternate in sign.…”

On the properties of fibotomic polynomials

“…However, Kitayama and Shiomi [2] showed that Ψ n (x) is often reducible in finite fields. More recently, Sagan and Tirrell [5] studied the bivariate Lucas polynomials and their factorization using Lucas atoms. The bivariate Lucas polynomials are defined such that L 1 (s, t) = 1, L 2 (s, t) = s, and L n (s, t) = L n−1 (s, t) • s + L n−2 (s, t) • t for all integers n ≥ 3.…”