2025/11/05 by Altizio, David
#26C10 #30B70 (Primary) 11R09 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2511.03847
The classical Stern sequence of positive integers was extended to a polynomial sequence Sn(λ) by Klavžar et. al. by defining S0(λ) = 0, S1(λ) = 1, and S2n(λ) = λSn(λ), S2n+1(λ) = Sn(λ) + Sn+1(λ). Dilcher et. al. conjectured that all roots of Sn(λ) lie in the half-plane \Re w < 1\. We make partial progress on this conjecture by proving that \|w-2| ≤ 1\⊆\mathbb C does not contain any roots of Sn(λ). Our proof uses the Parabola Theorem for convergence of complex continued fractions. As a corollary, we establish a conjecture of Ulas and Ulas by showing that Sp(λ) is irreducible in \mathbb Z[λ] whenever p is a positive prime.