A note on the arithmetic properties of Stern Polynomials

Abstract

We investigate the Stern polynomials defined by $B_0(t)=0,B_1(t)=1$, and for $n\ge 2$ by the recurrence relations $B_{2n}(t)=t{B}_n(t),$ $B_{2n+1}(t)=B_n(t)+B_{n+1}(t)$. We prove that all possible rational roots of that polynomials are $0,-1,-1/2,-1/3$. We give complete characterization of $n$ such that $deg(B_n)=deg(B_{n+1})$ and $deg(B_n)=deg(B_{n+1})=deg(B_{n+2})$. Moreover, we present some result concerning reciprocal Stern polynomials.

arXiv comment9 pages, submitted