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

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Published
2021-06-22
How to Cite
Frontczak, R., & Goy, T. (2021). General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function. Matematychni Studii, 55(2), 115-123. https://doi.org/10.30970/ms.55.2.115-123
Section
Articles