Further results on arctangent sums with applications to generalized Fibonacci numbers

Robert Frontczak
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 39–53
Full paper (PDF, 206 Kb)

Details

Authors and affiliations

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

Abstract

In this article, we extend a classical trigonometric addition formula for the arctangent function and derive new summation identities for Fibonacci and Lucas numbers. While most of the results seem to be new, we also recover some known expressions.

Keywords

• Arctangent sum
• Fibonacci number
• Lucas number

AMS Classification

• 11B37
• 11B39

References

1. Anglesio, J. (1993) Elementary problem 10292, Amer. Math. Monthly, Vol. 100, p. 291. Solution by several authors, Amer. Math. Monthly, Vol. 103, 1995, p. 270–272.
2. Adegoke, K. (2015) Infinite arctangent sums involving Fibonacci and Lucas numbers. Notes on Number Theory and Discrete Mathematics, 21(1), 56–66.
3. Boros, G. & Moll, V. H. (2005) Sums of arctangents and some formulas of Ramanujan, SCIENTIA Series A: Mathematical Sciences, 11, 13–24.
4. Borwein, J. M. & Borwein, P. B. (1987) Pi and the AGM, John Wiley & Sons, New York.
5. Frontczak, R. (2015) On Infinite Series Involving Fibonacci Numbers, International Journal of Contemporary Mathematical Sciences, 10(8), 363–379.
6. Frontczak, R. (2016) A Note on a Family of Alternating Fibonacci Sums. Notes on Number Theory and Discrete Mathematics, 22(2), 64–71.
7. Grimaldi, R. P. (2012) Fibonacci and Catalan numbers: An Introduction, John Wiley & Sons, New York.
8. Melham, R. S. (1999) Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica, 56(3), 309–317.
9. Melham, R. S. & Shannon, A. G. (1995) Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers, The Fibonacci Quarterly, 33(1), 32–40.
10. Berndt, B. C. (1985) Ramanujan’s Notebooks, Part I, Springer-Verlag, New York.
11. Rabinowitz, S. (1999) Algorithmic Summation of Reciprocals of Products of Fibonacci Numbers, The Fibonacci Quarterly, 37, 122–127.