TY - JOUR
AU - Frontczak, R.
AU - Goy, T.
PY - 2021/06/22
Y2 - 2021/10/16
TI - General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function
JF - Matematychni Studii
JA - Mat. Stud.
VL - 55
IS - 2
SE - Articles
DO - 10.30970/ms.55.2.115-123
UR - http://matstud.org.ua/ojs/index.php/matstud/article/view/128
SP - 115-123
AB - The purpose of this paper is to present closed forms for various types of infinite seriesinvolving 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 formulasfor 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.
ER -