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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2021-06-01
|
| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/128 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849415371151376384 |
|---|---|
| author | R. Frontczak T. Goy |
| author_facet | R. Frontczak T. Goy |
| author_sort | R. Frontczak |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d733c01bfb864841b5cd22bf39b02d92 |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2021-06-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-d733c01bfb864841b5cd22bf39b02d922025-08-20T03:33:32ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202021-06-0155211512310.30970/ms.55.2.115-123128General infinite series evaluations involving Fibonacci numbers and the Riemann zeta functionR. Frontczak0T. Goy1Landesbank Baden-W¨urttemberg (LBBW) Stuttgart, GermanyVasyl Stefanyk Precarpathian National UniversityThe 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/128fibonacci number; lucas number; riemann zeta function; digamma function; generating function |
| spellingShingle | R. Frontczak T. Goy General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function Математичні Студії fibonacci number; lucas number; riemann zeta function; digamma function; generating function |
| title | General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function |
| title_full | General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function |
| title_fullStr | General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function |
| title_full_unstemmed | General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function |
| title_short | General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function |
| title_sort | general infinite series evaluations involving fibonacci numbers and the riemann zeta function |
| topic | fibonacci number; lucas number; riemann zeta function; digamma function; generating function |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/128 |
| work_keys_str_mv | AT rfrontczak generalinfiniteseriesevaluationsinvolvingfibonaccinumbersandtheriemannzetafunction AT tgoy generalinfiniteseriesevaluationsinvolvingfibonaccinumbersandtheriemannzetafunction |