Continued fractions and polynomials related to hyperbinary representations
Volume 66 / 2018
Bulletin Polish Acad. Sci. Math. 66 (2018), 9-29
MSC: Primary 11B83; Secondary 11A55, 11B37, 11B75.
DOI: 10.4064/ba8126-12-2017
Published online: 22 January 2018
Abstract
Schinzel recently showed that the $n$th 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.
