Hyperbinary Expansions and Stern Polynomials

Karl Dilcher, Larry Ericksen


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.


Hyperbinary expansions; Stern sequence; Stern polynomials

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