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Continued fractions and polynomials related to hyperbinary representations

Volume 66 / 2018

Karl Dilcher, Larry Ericksen 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.

Authors

  • Karl DilcherDepartment of Mathematics and Statistics
    Dalhousie University
    Halifax, Nova Scotia, B3H 4R2, Canada
    e-mail
  • Larry EricksenP.O. Box 172
    Millville, NJ 08332-0172, U.S.A.
    e-mail

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