Generalized Stern polynomials and hyperbinary representations

• Autorzy: Dilcher K. , Ericksen L.

Identyfikatory

DOI

10.4064/ba8082-2-2017

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Stern sequence; Stern polynomials; hyperbinary expansions; generating functions

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 1
• Strony: 11--28
• Opis fizyczny: Bibliogr. 21 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] N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly 107(2000), 360-363.
• [3] L. Carlitz, Single variable Bell polynomials, Collect. Math. 14 (1960), 13-25.
• [4] L. Carlitz, A problem in partitions related to the Stirling numbers, Bull. Amer. Math. Soc. 70 (1964), 275-278.
• [5] K. Dilcher and L. Ericksen, Hyperbinary expansions and Stern polynomials, Electron. J. Combin. 22 (2015), no. 2, paper 2.24, 18 pp.
• [6] K. Dilcher and L. Ericksen, Some tilings, colorings and lattice paths via Stern polynomials, preprint, 2015.
• [7] K. Dilcher and L. Ericksen, Continued fractions and Stern polynomials, Ramanujan J., online (2017); doi:10.1007/s11139-016-9864-3.
• [8] K. Dilcher and K. B. Stolarsky, A polynomial analogue to the Stern sequence, Int. J. Number Theory 3 (2007), 85-103.
• [9] M. Gawron, A note on the arithmetic properties of Stern polynomials, Publ. Math. Debrecen 85 (2014), 453-465.
• [10] C. Giuli and R. Giuli, A primer on Stern's diatomic sequence, Part III: Additional results, Fibonacci Quart. 17 (1979), 318-320.
• [11] S. Klavžar, U. Milutinović, and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007), 86-95.
• [12] T. Mansour, q-Stern polynomials as numerators of continued fractions, Bull. Polish Acad. Sci. Math. 63 (2015), 11-18.
• [13] OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2011, http://oeis.org.
• [14] 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.
• [15] T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990.
• [16] A. Schinzel, On the factors of Stern polynomials (remarks on the preceding paper of M. Ulas), Publ. Math. Debrecen 79 (2011), 83-88.
• [17] A. Schinzel, Stern polynomials as numerators of continued fractions, Bull. Polish Acad. Sci. Math. 62 (2014), 23-27.
• [18] A. Schinzel, The leading coefficients of Stern polynomials, in: From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz (J. Sander et al., eds.), Springer, 2016, 427-434.
• [19] R. P. Stanley and H. S. Wilf, Refining the Stern diatomic sequence, preprint, 2010, http://www-math.mit.edu/~rstan/papers/stern.pdf.
• [20] M. Ulas, On certain arithmetic properties of Stern polynomials, Publ. Math. Debrecen 79 (2011), 55-81.
• [21] M. Ulas, Arithmetic properties of the sequence of degrees of Stern polynomials and related results, Int. J. Number Theory 8 (2012), 669-687.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).