General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function
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 ge...
| Published in: | Математичні Студії |
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| Main Authors: | , |
| Format: | Article |
| Language: | German |
| Published: |
Ivan Franko National University of Lviv
2021-06-01
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| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/128 |
| Summary: | 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. |
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| ISSN: | 1027-4634 2411-0620 |