Hyperbinary Expansions and Stern Polynomials

Abstract

We introduce an infinite class of polynomial sequences $a_t(n;z)$ with integer�parameter $t\geq 1$, which reduce to the well-known Stern (diatomic) sequence when $z=1$ and are $(0,1)$-polynomials when $t\geq 2$.�Using these polynomial sequences, we derive two different characterizations�of all hyperbinary expansions of an integer $n\geq 1$. Furthermore,�we study the polynomials $a_t(n;z)$ as objects in their own right,�obtaining a generating function and some consequences. We also prove results on the structure of these sequences, and determine�expressions for the degrees of the polynomials.