Generalized Fibonacci numbers and Bernoulli polynomials

Anthony G. Shannon, Ömür Deveci and Özgűr Erdağ
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 1, Pages 193—198
DOI: 10.7546/nntdm.2019.25.1.193-198
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Authors and affiliations

Anthony G. Shannon
Fellow, Warrane College, The University of New South Wales
Kensington NSW 2033, Australia

Ömür Deveci
Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey

Özgűr Erdağ
Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey

Abstract

Relationships, in terms of equations and congruences, are developed between the Bernoulli numbers and arbitrary order generalizations of the ordinary Fibonacci and Lucas numbers. Some of these are direct connections and others are analogous similarities.

Keywords

  • Fibonacci polynomials
  • Difference operators
  • Generalized Fibonacci and Lucas numbers
  • Bernoulli numbers and polynomials

2010 Mathematics Subject Classification

  • 11B39
  • 11B50
  • 11B68

References

  1. Carlitz, L. (1965). Recurrences for the Bernoulli and Euler numbers. Mathematische Nachrichten. 2 (3–4), 151–160.
  2. Hartree, D. R. (1958). Numerical Analysis. Oxford: Clarendon Press.
  3. Hoggatt, V. E. Jr, & Bicknell, M. (1973). Roots of Fibonacci polynomials. The Fibonacci Quarterly. 11 (3), 271–274.
  4. Hoggatt, V. E. Jr, & Long, C. T. (1974). Divisibility properties of generalized Fibonacci Polynomials. The Fibonacci Quarterly. 12 (2), 113–120.
  5. Knuth, D. E. (1992). Two notes on notation, American Mathematical Monthly, 99 (5), 403–422.
  6. Shannon, A. G. (1975). Fibonacci analogs of the classical polynomials. Mathematics Magazine. 48 (3), 123–130.
  7. Sun, Z.-W. (2003). General congruences for Bernoulli polynomials. Discrete Mathematics. 261 (1–3), 253–276.

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Cite this paper

APA

Shannon, A. G., Deveci, O. & Erdağ, Ö. (2019). Generalized Fibonacci numbers and Bernoulli polynomials. Notes on Number Theory and Discrete Mathematics, 25(1), 193-198, doi: 10.7546/nntdm.2019.25.1.193-198.

Chicago

Shannon, Anthony G. and Ömür Deveci and Özgűr Erdağ. “Generalized Fibonacci Numbers and Bernoulli Polynomials.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 193-198, doi: 10.7546/nntdm.2019.25.1.193-198.

MLA

Shannon, Anthony G. and Ömür Deveci and Özgűr Erdağ. “Generalized Fibonacci Numbers and Bernoulli Polynomials.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 193-198. Print, doi: 10.7546/nntdm.2019.25.1.193-198.

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