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SUM FORMULAE OF GENERALIZED FIBONACCI AND LUCAS NUMBERS

  • Cerin, Zvonko (Department of Mathematics, Zagreb University) ;
  • Bitim, Bahar Demirturk (Department of Mathematics, Sakarya University) ;
  • Keskin, Refik (Department of Mathematics, Sakarya University)
  • Received : 2017.12.28
  • Accepted : 2018.02.09
  • Published : 2018.03.25

Abstract

In this paper we obtain some formulae for several sums of generalized Fibonacci numbers $U_n$ and generalized Lucas numbers $V_n$ and their dual forms $G_n$ and $H_n$ by using extensions of an interesting identity by A. R. Amini for Fibonacci numbers to these four kinds of generalizations and their first and second derivatives.

Keywords

generalized Fibonacci sequence;generalized Lucas sequence;dual generalized Fibonacci sequence;dual generalized Lucas sequence;sum of products

References

  1. A.R.Amini, Fibonacci Numbers From a Long Division Formula, Mathematical Spectrum, 40 (2008), 59-61.
  2. H. Belbachir, F. Bencherif, Sums of Generalized Fibonacci and Lucas Numbers, arXiv: 0708.2347v1, 17.08.2007.
  3. Z. Cerin, Sums of Products of Generalized Fibonacci and Lucas Numbers, Demonstratio Mathematica, Vol. XLII (2) (2009), 247-258.
  4. P. Galister, Two Fibonacci Sums, A Variation, The Mathematical Gazette, 81 (1997), 85-88. https://doi.org/10.2307/3618776
  5. N. Gauthier, Fibonacci Sums of the Type ${\Sigma}r^mF_m$, The Mathematical Gazette, 79 (1995), 364-367. https://doi.org/10.2307/3618316
  6. A. F. Horadam, A Generalized Fibonacci sequence, American Mathematical Monthly, 68(5) (1961), 455-459. https://doi.org/10.1080/00029890.1961.11989696
  7. D. Kalman, R. Mena, The Fibonacci numbers-exposed, Mathematics Magazine, 76 (2003), 167-181.
  8. S. Rabinowitz, Algorithmic Manipulation of Fibonacci Identities, Applications of Fibonacci Numbers, Kluwer Academic Pub., Dordrect, The Netherlands, Vol. 6, 389-408, 1996.
  9. P. Ribenboim, The little book of big primes, Springer-Verlag, New York, 1991.
  10. P. Ribenboim, My numbers, My friends, Springer-Verlag Inc., New York, 2000.
  11. S. Vajda, Fibonacci and Lucas numbers and the golden section, Ellis Horwood Limited Publ., England, 1989.