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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


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Volume 64, Issue 4

Issues

On some new identities for the Fibonomial coefficients

Diego Marques / Pavel Trojovský
  • Department of Mathematics Faculty of Science, University of Hradec Králové, Rokitanského 62, Hradec Králové, Czech Republic
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Published Online: 2014-09-07 | DOI: https://doi.org/10.2478/s12175-014-0241-7

Abstract

Let F n be the nth Fibonacci number. The Fibonomial coefficients $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F$$ are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with $$\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1$$ and $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0$$. In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l.

MSC: Primary 11B39, 15A36; Secondary 15A15

Keywords: Fibonacci and Lucas numbers; Fibonomial coefficients; identity; sum

  • [1] FONTENÉ, G.: Généralisation d’ene formule connue, Nouv. Ann. Math. 4 (1915), 112. Google Scholar

  • [2] HORADAM, A. F.: Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J. 32 (1965), 437–446. http://dx.doi.org/10.1215/S0012-7094-65-03244-8CrossrefGoogle Scholar

  • [3] JARDEN, D.: Recurring Sequences: A Colletion of Papers, Riveon Lematematika, Jerusalem, 1966. Google Scholar

  • [4] LUCA, F.— MARQUES, D.— STĂNICĂ, P.: On the spacings between C-nomial coefficients, J. Number Theory 130 (2010), 82–100. http://dx.doi.org/10.1016/j.jnt.2009.07.015CrossrefWeb of ScienceGoogle Scholar

  • [5] MARQUES, D.— TOGBÉ, A.: Perfect powers among C-nomial coefficients, C. R. Math. Acad. Sci. Paris 348 (2010), 717–720. http://dx.doi.org/10.1016/j.crma.2010.06.006CrossrefGoogle Scholar

  • [6] MARQUES, D.— TROJOVSKÝ, P.: On some new sums of Fibonomial coefficients, Fibonacci Quart. 50 (2012), 155–163. Google Scholar

  • [7] RIORDAN, J.: Generating functions for powers of Fibonacci, Duke Math. J. 29 (1962), 5–12. http://dx.doi.org/10.1215/S0012-7094-62-02902-2CrossrefGoogle Scholar

  • [8] SHANNON, A. G.: A method of Carlitz applied to the k-th power generating function for Fibonacci numbers, Fibonacci Quart. 12 (1974), 293–299. Google Scholar

  • [9] SEIBERT, J.— TROJOVSKÝ, P.: On some identities for the Fibonomial coefficients, Math. Slovaca 55 (2005), 9–19. Web of ScienceGoogle Scholar

  • [10] TORRETTO, R.— FUCHS, A.: Generalized binomial coefficients, Fibonacci Quart. 2 (1964), 296–302. Google Scholar

  • [11] VAJDA, S.: Fibonacci and Lucas Numbers, and the Golden Section, John Wiley and Sons, New York, 1989. Google Scholar

  • [12] VOROBIEV, N. N.: Fibonacci Numbers, Birkhäuser, Basel, 2003. Google Scholar

  • [13] WARD, M.: A calculus of sequences, Amer. J. Math. 58 (1936), 255–266. http://dx.doi.org/10.2307/2371035CrossrefGoogle Scholar

  • [14] SLOANE, N. J. A.: The On-Line Encyclopedia of Integer Sequences (OEIS). http://oeis.org/. Google Scholar

About the article

Published Online: 2014-09-07

Published in Print: 2014-08-01


Citation Information: Mathematica Slovaca, Volume 64, Issue 4, Pages 809–818, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-014-0241-7.

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© 2014 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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