Strong arithmetic property of certain Stern polynomials

Abstract: Let $B_{n}(t)$ be the $n$th Stern polynomial, i.e., the $n$th term of the sequence defined recursively as $B_{0}(t)=0, B_{1}(t)=1$ and $B_{2n}(t)=tB_{n}(t), B_{2n+1}(t)=B_{n}(t)+B_{n-1}(t)$ for $n\in\N$. It is well know that $i$th coefficient in the polynomial $B_{n}(t)$ counts the number of hyperbinary representations of $n-1$ containing exactly $i$ digits 1. In this note we investigate the existence of odd solutions of the congruence \begin{equation*} B_{n}(t)\equiv 1+rt\frac{t^{e(n)}-1}{t-1}\pmod{m}, \end{equation*} where $m\in\N_{\geq 2}$ and $r\in{0,\ldots,m-1}$ are fixed and $e(n)=\op{deg}B_{n}(t)$. We prove that for $m=2$ and $r\in{0,1}$ and for $m=3$ and $r=0$, there are infinitely many odd numbers $n$ satisfying the above congruence. We also present results of some numerical computations.