Closed-form evaluations of Fibonacci–Lucas reciprocal sums with three factors

Robert Frontczak
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 2, Pages 104—116
Download full paper: PDF, 189 Kb

Details

Authors and affiliations

Robert Frontczak
Landesbank Baden-Wuerttemberg
Am Hauptbahnhof 2, 70173 Stuttgart, Germany

Abstract

In this article we present expressions for certain types of reciprocal Fibonacci and Lucas sums. The common feature of the sums is that in each case the denominator of the summand consists of a product of three Fibonacci or Lucas numbers.

Keywords

  • Fibonacci number
  • Lucas number
  • Reciprocal sum

AMS Classification

  • 11B37
  • 11B39

References

  1. Almkvist, G. (1986) A Solution to a Tantalizing Problem, The Fibonacci Quarterly, 24 (4), 316–322.
  2. Andr-Jeannin, R. (1990) Lambert Series and the Summation of Reciprocals of Certain Fibonacci-Lucas-Type Sequences, The Fibonacci Quarterly, Vol. 28 (3), 223–226.
  3. Backstrom, R.P (1981)On Reciprocal Series Related to Fibonacci Numbers with Subscripts in Arithmetic Progression, The Fibonacci Quarterly, 19, 14–21.
  4. Borwein, J.M. & Borwein, P.B. (1987) Pi and the AGM, John Wiley & Sons, New York.
  5. Brousseau, B.A. (1969) Summation of Infinite Fibonacci Series, The Fibonacci Quarterly, 7 (2), 143–168.
  6. Carlitz, L. (1971) Reduction Formulas for Fibonacci Summations, The Fibonacci Quarterly, 9(5), 449–466, 510–511.
  7. Frontczak, R. (2016) A Note on a Family of Alternating Fibonacci Sums, Notes on Number Theory and Discrete Mathematics 22(2), 64–71.
  8. Frontczak, R. (2017) Further Results on Arctangent Sums with Applications to Generalized Fibonacci Numbers, Notes on Number Theory and Discrete Mathematics, 23(1), 39–53.
  9. Frontczak, R. (2017) Summation of Some Finite Fibonacci Lucas Series, unpublished manuscript.
  10. Good, I. J., (1974) A reciprocal series of Fibonacci numbers, The Fibonacci Quarterly, 12, 346.
  11. Horadam, A. F. (1988) Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, 26(2), 98–114.
  12. Hu, H., Sun, Z.-W., & Liu, J.-X. (2001) Reciprocal sums of second-order recurrent sequences, The Fibonacci Quarterly, 39, 214–220.
  13. Melham, R. S. (1999) Lambert Series and Elliptic Functions and Certain Reciprocal Sums, The Fibonacci Quarterly, 37, 208–212.
  14. Melham, R. S. (2000) Summation of reciprocals which involve products of terms from generalized Fibonacci sequences, The Fibonacci Quarterly, 38, 294–298.
  15. Melham, R. S. (2001) Summation of reciprocals which involve products of terms from generalized Fibonacci sequences – Part II, The Fibonacci Quarterly, 39, 264–267.
  16. Melham, R. S. (2002) Reduction Formulas for the Summation of Reciprocals in Certain Second-Order Recurring Sequences, The Fibonacci Quarterly, 40(1), 71–75.
  17. Melham, R. S. (2003) On Some Reciprocal Sums of Brousseau: An Alternative Approach to that of Carlitz, The Fibonacci Quarterly, 41(1), 59–62.
  18. Melham, R. S. (2012) On finite sums of Good and Shar that involve reciprocals of Fibonacci numbers, Integers, Vol. 12 (A61).
  19. Melham, R. S.(2013) Finite Sums That Involve Reciprocals of Products of Generalized Fibonacci Numbers, Integers, Vol. 13 (A40).
  20. Melham R. S. (2014) More on Finite Sums that Involve Reciprocals of Products of Generalized Fibonacci Numbers, Integers, Vol. 14 (A4).
  21. Melham, R. S. (2015) On Certain Families of Finite Reciprocal Sums that Involve Generalized Fibonacci Numbers, The Fibonacci Quarterly, 53(4), 323–334.
  22. Melham, R. S., & Shannon, A. G. (1995) On Reciprocal Sums of Chebyshev Related Sequences, The Fibonacci Quarterly, 33(3), 194–202.
  23. Rabinowitz, S. (1999) Algorithmic Summation of Reciprocals of Products of Fibonacci Numbers, The Fibonacci Quarterly, 37, 122–127.

Related papers

Cite this paper

Frontczak, R. (2017). Closed-form evaluations of Fibonacci–Lucas reciprocal sums with three factors. Notes on Number Theory and Discrete Mathematics, 23(2), 104—116.

Comments are closed.