Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients $\binom{2n}{n}$

TL;DR

This paper derives new series involving harmonic numbers, Fibonacci numbers, and central binomial coefficients using a straightforward transformation, providing novel evaluations including a series for er(2) that differ from traditional hypergeometric methods.

Contribution

It introduces a simple transformation method to find series involving Fibonacci numbers and harmonic numbers, offering new evaluations and a novel series for er(2).

Findings

New series involving central binomial coefficients and Fibonacci numbers.

A novel series representation for er(2).

Alternative approach to hypergeometric function methods.

Abstract

We find various series that involves the central binomial coefficients $\binom{2n}{n}$, harmonic numbers and Fibonacci Numbers.\\ Contrary to the traditional hypergeometric function $_pF_q$ approach, our method utilizes a straightforward transformation to obtain new evaluations linked to Fibonacci numbers and the golden ratio. Before the end of this paper, we also gave a new series representation for $\zeta(2)$.