Abstract
In this paper, the authors investigate two special families of series involving the reciprocal central binomial coefficients and Lucas numbers. Connections with several familiar sums representing the integer-valued Riemann zeta function are also pointed out.
Highlights
Introduction and MotivationOne of the important and widely and extensively investigated functions in Analytic Number Theory is the Riemann Zeta function ζ(s), which is defined for (s) > 1 by ∞ ∑ k=1 1 ks = 1 1 − 2−s (2k 1 −1)s (s) > 1 ζ(s) := (1)1 − 21−s (−1)k−1 ks (s) > 0; s = 1 and for (s) 1; s = 1 by its meromorphic continuation
The computation of infinite series containing reciprocal central binomial coefficients is a challenging issue and it is currently an active field in number theory and experimental mathematics. The interest in these series comes from the existence of connections to certain generating functions of the zeta function, special zeta values and other important constants such as the golden ratio
Before stating our first main result, we recall some basic facts about Fibonacci and Lucas numbers
Summary
One of the important and widely and extensively investigated functions in Analytic Number Theory is the Riemann Zeta function ζ(s), which is defined for (s) > 1 by. The computation of infinite series containing reciprocal central binomial coefficients is a challenging issue and it is currently an active field in number theory and experimental mathematics. The interest in these series comes from the existence of connections to certain generating functions of the zeta function, special zeta values and other important constants such as the golden ratio. The goal of this paper is to study two families of Apéry-like series with coefficients involving Fibonacci (Lucas) numbers. One such family of series is evaluated here in closed form.
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