Zeros of Stern polynomials in the complex plane

The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(\lambda )$ by Klavžar et. al. by defining $S_0(\lambda )=0$, $S_1(\lambda )=1$, and $$S_{2n}(\lambda )=\lambda S_n(\lambda ),S_{2n+1}(\lambda )=S_n(\lambda )+S_{n+1}(\lambda )\mathrm{.}$$ Dilcher et. al. conjectured that all roots of $S_n(\lambda )$ lie in the half-plane $\{\mathrm{Re}w<1\}$. We make partial progress on this conjecture by proving that $\{\mathrm{\mid }w-2\mathrm{\mid }\le 1\}\subseteq \mathbb{C}$ 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.