Another generalization of the Fibonacci and Lucas numbers

A. G. Shannon
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 16, 2010, Number 3, Pages 11—17
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Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales,
PO Box 123, Kensington, NSW 1465, Australia

Abstract

This paper considers some generalizations of the Fibonacci and Lucas numbers which are essentially ratios of the former, and hence not necessarily integers. Nevertheless, some new and elegant results emerge as well as variations on well-established identities.

Keywords

  • Fibonacci numbers
  • Fundamental Lucas numbers
  • Primordial Lucas Numbers
  • Simson’s identity

AMS Classification

  • 11B39
  • 97F60

References

    1. Bérczes, Attila, Kálmán, Istwán Pink. 2010. On Generalized Balancing Sequences. The Fibonacci Quarterly. 48: 121-128.
    2. Barakat, Richard. 1964. The Matrix Operator eX and the Lucas Polynomials. Journal of Mathematics and Physics. 43: 332-335.
    3. Carlitz, L. 1960. Congruence Properties of Certain Polynomial Sequences. Acta Arithmetica. 6: 149-158.
    4. Dickinson, David. 1950. On Sums Involving Binomial Coefficients. American Mathematical Monthly. 57: 82-86.
    5. Horadam, A.F. 1965. Generating Functions for Powers of a Certain Generalized Sequence of Numbers. Duke Mathematical Journal. 32: 437-446.
    6. Jerbic, Stephen K. 1968. Fibonomial Coefficients. Master of Arts Thesis, San Jose State College, California.
    7. Lucas, Edouard. 1878. Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics. 1: 184-240, 289-321.
    8. Shannon, A.G. 1972. Iterative Formulas Associated with Third Order Recurrence Relations. S.I.A.M. Journal on Applied Mathematics. 23: 364-368.
    9. Shannon, A.G. 1974. Advanced Problem H-233. The Fibonacci Quarterly. 12: 108.
    10. Whitney, Raymond E. 1970. On a Class of Difference Equations. The Fibonacci Quarterly. 8: 470-475.

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      Cite this paper

      APA

      Shannon, A. G. (2010). Another generalization of the Fibonacci and Lucas numbers. Notes on Number Theory and Discrete Mathematics, 16(3), 11-17.

      Chicago

      Shannon, AG. “Another Generalization of the Fibonacci and Lucas Numbers.” Notes on Number Theory and Discrete Mathematics 16, no. 3 (2010): 11-17.

      MLA

      Shannon, AG. “Another Generalization of the Fibonacci and Lucas Numbers.” Notes on Number Theory and Discrete Mathematics 16.3 (2010): 11-17. Print.

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