两个关于 $1/\pi$ 的 Ramanujan 级数的推广
New Generalizations of Two Ramanujan Series for $1/π$
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中文总结 AI 辅助
利用超几何求和恒等式，将两个 Ramanujan 的 $1/\pi$ 级数推广为无穷族，并给出含调和数及奇调和数的对应族。
中文摘要 AI 辅助
通过利用两个已知的超几何求和恒等式，我们将两个著名的关于 $1/\pi$ 的 Ramanujan 级数推广为无穷级数族中的成员。我们还找到了涉及调和数和奇调和数的相应级数族。
英文摘要
By utilizing two known hypergeometric summation identities, we extend two \mbox{well-known} Ramanujan series for $1/π$ by making each of them a member of an infinite family of series. We also find the corresponding families involving harmonic numbers and odd harmonic numbers. To further illustrate our method we derive families of \mbox{Ramanujan-like} series for $1/π$ associated with a hypergeometric series derived by Lavoie, two hypergeometric series from Bailey's book and a recent hypergeometric series found by Campbell. Thus, by leveraging known hypergeometric summation identities, our approach bypasses the more tedious, traditional limits or complex Fourier-Legendre expansions; offering a cleaner, unified template for generating Ramanujan-like series for $1/π$.
发表机构
- Obafemi Awolowo University（奥巴费米·阿沃洛沃大学）
机构由 AI 辅助整理，请以论文原文为准。
