On Some Series Involving the Central Binomial Coefficients

TL;DR

This paper investigates series involving central binomial coefficients, deriving new closed-form formulas, analyzing convergence, and establishing identities with Fibonacci and Lucas numbers.

Contribution

It introduces novel series representations and identities involving central binomial coefficients, including those with alternating signs and connections to Fibonacci and Lucas numbers.

Findings

Derived new closed-form series representations

Analyzed convergence properties of alternating series

Established identities linking binomial coefficients with Fibonacci and Lucas numbers

Abstract

In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and examine the convergence properties of infinite series with a repeating alternation pattern of signs involving central binomial coefficients. More concretely, we derive the series $$\sum\limits_{n=0}^{\infty}\frac{(-1)^{\omega_n}}{2n+1}\tbinom{2n}{n}x^n,\,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{\omega_n}}\tbinom{2n}{n}x^n\,\,\, \text{and} \,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{\omega_n}}n\tbinom{2n}{n}x^n,$$ where $\omega_n$ represents both $\lfloor\frac{n}{2}\rfloor$ and $\lceil\frac{n}{2}\rceil$. Also, we present novel series involving Fibonacci and Lucas numbers, deriving many interesting identities.