On Some Series Involving the Central Binomial Coefficients
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915291463680000 |
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| author | Adegoke, Kunle Frontczak, Robert Goy, Taras |
| author_facet | Adegoke, Kunle Frontczak, Robert Goy, Taras |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Some Series Involving the Central Binomial Coefficients Adegoke, Kunle Frontczak, Robert Goy, Taras Combinatorics 40A30, 11B37, 11B39 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. |
| title | On Some Series Involving the Central Binomial Coefficients |
| topic | Combinatorics 40A30, 11B37, 11B39 |
| url | https://arxiv.org/abs/2505.11575 |