Generalized Stern polynomials and hyperbinary representations

Volume 65 / 2017

Karl Dilcher, Larry Ericksen 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


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.


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

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