On Some Series Involving the Central Binomial Coefficients
Kunle Adegoke, Robert Frontczak, Taras Goy
TL;DR
The paper analyzes infinite series built from central binomial coefficients with sign patterns determined by floor/ceiling half-indexing, deriving closed-form representations and convergence properties. Using a fundamental lemma on the real and imaginary parts of $\arcsin((1+i)x)$, it obtains Maclaurin expansions that yield identities expressed through $\arctan$, $\arctanh$, $\arccot$, and $\arccoth$, plus numerous special cases and trig/hyperbolic variants. It then connects these central-binomial sums to Fibonacci and Lucas numbers via Binet’s formulas, producing many explicit identities in terms of the golden ratio and related functions, with a broad set of examples and variants. The work broadens the toolkit for evaluating alternating central-binomial series and suggests avenues to extend the approach to higher-order binomial coefficients and additional special-function links.
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 $$\sum\limits_{n=0}^{\infty}\frac{(-1)^{ω_n}}{2n+1}\tbinom{2n}{n}x^n,\,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}\tbinom{2n}{n}x^n\,\,\, \text{and} \,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{ω_n}}n\tbinom{2n}{n}x^n,$$ where $ω_n$ 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.