On Some Series Involving the Central Binomial Coefficients

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 $$\sum _{n=0}^{\infty }\frac{(-1{)}^{\ddot{I}{‰}_n}}{2n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)x^n,\sum _{n=0}^{\infty }(-1{)}^{\ddot{I}{‰}_n}(\genfrac{}{}{0ex}{}{2n}{n})x^n\text{and}\sum _{n=0}^{\infty }(-1{)}^{\ddot{I}{‰}_n}n\left(\genfrac{}{}{0ex}{}{2n}{n}\right)x^n,$$ where $\ddot{I}{‰}_n$ 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.

15 pages