Continued fractions and polynomials related to hyperbinary representations

• Autorzy: Dilcher K. , Ericksen L.

Identyfikatory

DOI

10.4064/ba8126-12-2017

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Stern sequence; Stern polynomials; hyperbinary expansions; continued fractions; recursion

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 1
• Strony: 9--29
• Opis fizyczny: Bibliogr. 15 poz., tab.

Twórcy

autor

Dilcher K.

• Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada

autor

Ericksen L.

• P.O. Box 172, Millville, NJ 08332-0172, U.S.A.

Bibliografia

• [1] B. Bates and T. Mansour, The q-Calkin-Wilf tree, J. Combin. Theory Ser. A 118 (2011), 1143-1151.
• [2] K. Dilcher and L. Ericksen, Hyperbinary expansions and Stern polynomials, Electron. J. Combin. 22 (2015), paper 2.24, 18 pp.
• [3] K. Dilcher and L. Ericksen, Continued fractions and Stern polynomials, Ramanujan J. 45 (2018), 659-681.
• [4] K. Dilcher and L. Ericksen, Generalized Stern polynomials and hyperbinary representations, Bull. Polish Acad. Sci. Math. 65 (2017), 11-28.
• [5] K. Dilcher and K. B. Stolarsky, A polynomial analogue to the Stern sequence, Int. J. Number Theory 3 (2007), 85-103.
• [6] M. Gawron, A note on the arithmetic properties of Stern polynomials, Publ. Math. Debrecen 85 (2014), 453-465.
• [7] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, Reading, MA, 1994.
• [8] S. Klavžar, U. Milutinović, and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007), 86-95.
• [9] T. Mansour, q-Stern polynomials as numerators of continued fractions, Bull. Polish Acad. Sci. Math. 63 (2015), 11-18.
• [10] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2011, http://oeis.org.
• [11] B. Reznick, Some binary partition functions, in: Analytic Number Theory: Proceedings of a Conference in Honor of Paul T. Bateman (B. C. Berndt et al., eds.), Birkhäuser, Boston, 1990, 451-477.
• [12] A. Schinzel, On the factors of Stern polynomials (remarks on the preceding paper of M. Ulas), Publ. Math. Debrecen 79 (2011), 83-88.
• [13] A. Schinzel, Stern polynomials as numerators of continued fractions, Bull. Polish Acad. Sci. Math. 62 (2014), 23-27.
• [14] R. P. Stanley and H. S. Wilf, Rening the Stern diatomic sequence, preprint, 2010, http://www-math.mit.edu/~rstan/papers/stern.pdf.
• [15] M. Ulas, On certain arithmetic properties of Stern polynomials, Publ. Math. Debrecen 79 (2011), 55-81.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).