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.
@article{10_37236_4822,
author = {Karl Dilcher and Larry Ericksen},
title = {Hyperbinary expansions and {Stern} polynomials},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {2},
doi = {10.37236/4822},
zbl = {1312.05016},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4822/}
}
TY - JOUR
AU - Karl Dilcher
AU - Larry Ericksen
TI - Hyperbinary expansions and Stern polynomials
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/4822/
DO - 10.37236/4822
ID - 10_37236_4822
ER -
%0 Journal Article
%A Karl Dilcher
%A Larry Ericksen
%T Hyperbinary expansions and Stern polynomials
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/4822/
%R 10.37236/4822
%F 10_37236_4822
Karl Dilcher; Larry Ericksen. Hyperbinary expansions and Stern polynomials. The electronic journal of combinatorics, Tome 22 (2015) no. 2. doi: 10.37236/4822
