Rad HAZU, Matematičke znanosti, Vol. 19 (2015), 1-12.

FORMULAS FOR QUADRATIC SUMS THAT INVOLVE GENERALIZED FIBONACCI AND LUCAS NUMBERS

Zvonko Čerin

Kopernikova 7, 10020 Zagreb, Croatia
e-mail: cerin@math.hr


Abstract.   We improve on Melham’s formulas in [10, Section 4] for certain classes of finite sums that involve generalized Fibonacci and Lucas numbers. Here we study the quadratic sums where products of two of these numbers appear. Our results show that most of his formulas are the initial terms of a series of formulas, that the analogous and somewhat simpler identities hold for associated dual numbers and that besides the alternation according to the numbers (-1)n(n+1)/2 it is possible to get similar formulas for the alternation according to the numbers (-1)n(n-1)/2. We also consider twelve quadratic sums with binomial coefficients that are products.

2010 Mathematics Subject Classification.   11B39, 11Y55, 05A19.

Key words and phrases.   (generalized) Fibonacci number, (generalized) Lucas number, factor, sum, alternating, binomial coefficient, product.


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References:

  1. G. E. Bergum and V. E. Hoggatt, Jr., Sums and products for recurring sequences, Fibonacci Quart. 13 (1975), 115-120.
    MathSciNet

  2. Z. Čerin, On factors of sums of consecutive Fibonacci and Lucas numbers, Ann. Math. Inform. 41 (2013), 19-25.
    MathSciNet

  3. Z. Čerin, Sums of products of generalized Fibonacci and Lucas numbers, Demonstratio Math. 42 (2009), 247-258.
    MathSciNet

  4. Z. Čerin, Formulas for linear sums that involve generalized Fibonacci and Lucas numbers, Sarajevo J. Math. 11 (2015), 3-15.
    MathSciNet     CrossRef

  5. H. Freitag, On summations and expansions of Fibonacci numbers, Fibonacci Quart. 11 (1973), 63-71.
    MathSciNet

  6. A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), 161-176.
    MathSciNet

  7. C. T. Long, Some binomial Fibonacci identities, in: Applications of Fibonacci Numbers 3 (eds. G. E. Bergum et al.), Kluwer, Dordrecht, 1990, pp. 241-254.
    MathSciNet

  8. E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics 1 (1878), 184-240.
    MathSciNet     CrossRef

  9. R. S. Melham, Summation of reciprocals which involve products of terms from generalized Fibonacci sequences, Fibonacci Quart. 38 (2000), 294-298.
    MathSciNet

  10. R. S. Melham, Certain classes of finite sums that involve generalized Fibonacci and Lucas numbers, Fibonacci Quart. 42 (2004), 47-54.
    MathSciNet

  11. D. L. Russell, Summation of second-order recurrence terms and their squares, Fibonacci Quart. 19 (1981), 336-340.
    MathSciNet

  12. D. L. Russell, Notes on sums of products of generalized Fibonacci numbers, Fibonacci Quart. 20 (1982), 114-117.
    MathSciNet

  13. N. Sloane, On-Line Encyclopedia of Integer Sequences, http://oeis.org/.

  14. S. Vajda, Fibonacci & Lucas Numbers and the Golden Section: Theory and Applications, Ellis Horwood Limited, Chichester, 1989.
    MathSciNet

  15. C. R. Wall, Some congruences involving generalized Fibonacci numbers, Fibonacci Quart. 17 (1979), 29-33.
    MathSciNet


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