Infinite arctangent sums involving Fibonacci and Lucas numbers

Kunle Adegoke
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 1, Pages 56–66
Full paper (PDF, 150 Kb)

Details

Authors and affiliations

Kunle Adegoke
Department of Physics, Obafemi Awolowo University
Ile-Ife, 220005 Nigeria

Abstract

We derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of ‘celebrated’ results appear as particular cases of the more general formulas derived here.

Keywords

• Fibonacci numbers
• Lucas numbers
• Lehmer’s formula
• Arctangent sums
• Infinite sums

AMS Classification

• 11B39
• 11Y60

References

1. Bragg, L. (2001) Arctangent sums. The College Mathematics Journal, 32(4), 255–257.
2. Hayashi, K. (2003) Fibonacci numbers and the arctangent function. Mathematics Magazine, 76(3), 214–215.
3. Hoggatt Jr, V. E., & Ruggles, I. D. (1964) A primer for the Fibonacci numbers: Part V. The Fibonacci Quarterly, 2(1), 46–51.
4. Mahon, J. M., Br., & Horadam, A. F. (1985) Inverse trigonometrical summation formulas involving Pell polynomials. The Fibonacci Quarterly, 23(4), 319–324.
5. Melham, R. S. & Shannon, A. G. (1995) Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers. The Fibonacci Quarterly, 33(1),32–40.
6. Basin, S. L. & Hoggatt Jr., V. E. (1964) A primer for the Fibonacci numbers: Part I. The Fibonacci Quarterly, 2(1), 13–17.
7. Dunlap, R. A. (2003) The Golden Ratio and Fibonacci Numbers. World Scientific.
8. Howard, F. T. (2003) The sum of the squares of two generalized Fibonacci numbers. The Fibonacci Quarterly, 41(1), 80–84.