On Some Series Involving the Central Binomial Coefficients

2025/05/16 by Kunle Adegoke, Robert Frontczak, Adegoke, Kunle +3
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2505.11575

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 ∑_n=0^∞\frac(-1)^ω_n2n+1\tbinom2nnxⁿ, ∑_n=0^∞(-1)^ω_n\tbinom2nnxⁿ and ∑_n=0^∞(-1)^ω_nn\tbinom2nnxⁿ, where ω_n represents both \lfloor(n)/(2)\rfloor and \lceil(n)/(2)\rceil. Also, we present novel series involving Fibonacci and Lucas numbers, deriving many interesting identities.

Citations

• Evaluation of some alternating series involving the binomial coefficients C(4n,2n)
• Fibonacci-Catalan Series
• On a family of infinite series with reciprocal Catalan numbers
• Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations
• The On-Line Encyclopedia of Integer Sequences

Related

• Explicit Identities and new results for Infinite Series associated with the Ratio of Central Binomial Coefficients
• Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients \binom2nn
• Evaluation of some alternating series involving the binomial coefficients C(4n,2n)
• A Generalized Apery Series
• Sums of series involving central binomial coefficients & harmonic numbers