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2 Bulletin of the Polish Academy of Sciences. Mathematics

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Continued fractions and polynomials related to hyperbinary representations

Dilcher K., Ericksen L.

Bulletin of the Polish Academy of Sciences. Mathematics

2018

Vol. 66, no. 1

9--29

Schinzel recently showed that the nth Stern polynomial of Klavžar et al. is the numerator of a certain finite continued fraction. This was subsequently extended by Mansour to q-Stern polynomials. We extend these results further to a 2-parameter bivariate analogue of the sequence of Stern polynomials which arise naturally in the characterization of hyperbinary representations of a given integer. In the process we define a class of companion polynomials with which we can determine the denominators of the continued fractions in question.

Generalized Stern polynomials and hyperbinary representations

Dilcher K., Ericksen L.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 1

11--28

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 coeffcients. 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.