Generalized Stern polynomials and hyperbinary representations
Volume 65 / 2017
Bulletin Polish Acad. Sci. Math. 65 (2017), 11-28
MSC: Primary 05A15; Secondary 11B83.
DOI: 10.4064/ba8082-2-2017
Published online: 17 March 2017
Abstract
We use two different but related types of generalized Stern polynomials, recently introduced by the authors, to give complete characterizations of all hyperbinary expansions of a given positive integer. We also derive explicit formulas for these generalized Stern polynomials and use them to establish further characterizations of hyperbinary expansions, using binomial coefficients. We then introduce a 2-parameter analogue of the two types of polynomials, which leads to more explicit versions of earlier results. Finally, we explore further generalizations of the polynomials studied in this paper.
