Zeros of Stern polynomials in the complex plane

Abstract

The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(\lambda)$ by Klav\v{z}ar et. al. by defining $S_0(\lambda) = 0$, $S_1(\lambda) = 1$, and $$S_{2n}(\lambda) = \lambda S_n(\lambda),\quad S_{2n+1}(\lambda) = S_n(\lambda) + S_{n+1}(\lambda).$$ Dilcher et. al. conjectured that all roots of $S_n(\lambda)$ lie in the half-plane $\{\operatorname{Re} w < 1\}$. We make partial progress on this conjecture by proving that $\{|w-2| \leq 1\}\subseteq\mathbb C$ does not contain any roots of $S_n(\lambda)$. 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 $S_p(\lambda)$ is irreducible in $\mathbb Z[\lambda]$ whenever $p$ is a positive prime.