A note on the coefficient array of a generalized Fibonacci polynomial

A. G. Shannon and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310-5132, Online ISSN 2367-8275
Volume 26, 2020, Number 4, Pages 206—212
DOI: 10.7546/nntdm.2020.26.4.206-212
Download full paper: PDF, 156 Kb

Details

Authors and affiliations

A. G. Shannon
Warrane College, the University of New South Wales
Kensington, NSW 2033, Australia

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

Abstract

A particular version of a Fibonacci polynomial is presented and the coefficients are tabulated to bring out some of their number theory properties with known results. Generalizations of the fundamental and primordial Lucas sequences are used in the proofs.

Keywords

  • Fibonacci numbers and polynomials
  • Lucas numbers
  • Bernoulli polynomials
  • Falling factorial coefficients
  • Umbral calculus.

2010 Mathematics Subject Classification

  • 11B39
  • 11B68

References

  1. Borel, É. (1899). Mémoire sur les séries divergentes, Annales scientifiques de l’École Normale Supérieure. Serie 3, 16(1), 9–131.
  2. Hardy, G. H., & Ramanujan, S. (1917). Asymptotic formulae in combinatory analysis.
    Proceedings of the London Mathematical Society. Series 2, 16(1), 75–115.
  3. Hoggatt, V. E. Jr. (1969). Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin. [4] Hoggatt, V. E, Jr., & Bicknell, M. (1973). Roots of Fibonacci polynomials. The Fibonacci Quarterly. 11(3), 271–274.
  4. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly. 3(3), 161–176.
  5. Lucas, E. (1878). Théorie des Fonctions Numériques Simplement Périodiques. American Journal of Mathematics. 1, 184–240.
  6. Rota, G.-C., Kahaner, D., & Oslyzko, A. (1975). Finite Operator Calculus. New York: Academic Press.
  7. Shannon, A. G. (1975). Fibonacci analogs of the classical polynomials. Mathematics Magazine. 48(3), 123–130.
  8. Shannon, A. G., & Deveci, O. (2018). Some generalized Fibonacci and Hermite polynomials. JP Journal of Algebra, Number Theory and Applications. 40(4), 419–427.
  9. Singh, D. (1952). The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers. Mathematics Student. 20(1), 66–70.
  10. Sloane, N. J. A. (1973). A Handbook of Integer Sequences. New York: Academic Press.

Related papers

Cite this paper

Shannon, A. G., & Deveci, Ö. (2020). A note on the coefficient array of a generalized Fibonacci polynomial. Notes on Number Theory and Discrete Mathematics, 26 (4), 206-212, doi: 10.7546/nntdm.2020.26.4.206-212.

Comments are closed.