General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function

Keywords: Fibonacci number; Lucas number; Riemann zeta function; digamma function; generating function

Abstract

The purpose of this paper is to present closed forms for various types of infinite series
involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
To prove our results, we will apply some conventional arguments and combine the Binet formulas
for these sequences with generating functions involving the Riemann zeta function and some known series evaluations.
Among the results derived in this paper, we will establish that

$\displaystyle
\sum_{k=1}^\infty (\zeta(2k+1)-1) F_{2k} = \frac{1}{2},\quad
\sum_{k=1}^\infty (\zeta(2k+1)-1) \frac{L_{2k+1}}{2k+1} = 1-\gamma,$


where $\gamma$ is the familiar Euler-Mascheroni constant.

References

M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1972.

T. Arakawa, T. Ibukiyama, M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, 2014.

J.M. Borwein, D.M. Bradley, R.E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math., 121 (2000), 247–296.

H.M. Edwards, Riemann’s Zeta Function, Academic Press, 1974.

R. Frontczak, Problem B-1267, Fibonacci Quart., 58 (2020), №2, 179.

R. Frontczak, Problem H-859, Fibonacci Quart., 58 (2020), №3, 281.

R. Frontczak, Infinite series involving Fibonacci numbers and the Riemann zeta function, Notes Number Theory Discrete Math., 26 (2020), №2, 159–166. doi: 10.7546/nntdm.2020.26.2.159-166

T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2017.

A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series. Vol. 1. Elementary Functions, Gordon & Breach Sci. Publ., 1986.

H.M. Srivastava, J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, 2012.

N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences, https://oeis.org.

Published
2021-06-22
How to Cite
1.
Frontczak R, Goy T. General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function. Mat. Stud. [Internet]. 2021Jun.22 [cited 2021Sep.17];55(2):115-23. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/128
Section
Articles