拉马努金型系列,重访

Pub Date : 2023-10-05 DOI:10.4153/s0008439523000772
Dongxi Ye
{"title":"拉马努金型系列,重访","authors":"Dongxi Ye","doi":"10.4153/s0008439523000772","DOIUrl":null,"url":null,"abstract":"Abstract In this note, we revisit Ramanujan-type series for $\\frac {1}{\\pi }$ and show how they arise from genus zero subgroups of $\\mathrm {SL}_{2}(\\mathbb {R})$ that are commensurable with $\\mathrm {SL}_{2}(\\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\\frac {1}{\\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\\frac {1}{\\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ramanujan type series for , revisited\",\"authors\":\"Dongxi Ye\",\"doi\":\"10.4153/s0008439523000772\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In this note, we revisit Ramanujan-type series for $\\\\frac {1}{\\\\pi }$ and show how they arise from genus zero subgroups of $\\\\mathrm {SL}_{2}(\\\\mathbb {R})$ that are commensurable with $\\\\mathrm {SL}_{2}(\\\\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\\\\frac {1}{\\\\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\\\\frac {1}{\\\\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-10-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4153/s0008439523000772\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008439523000772","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

在本文中,我们重新讨论$\frac {1}{\pi }$的ramanujan型级数,并说明它们是如何从与$\mathrm {SL}_{2}(\mathbb {Z})$可通约的$\mathrm {SL}_{2}(\mathbb {R})$的属零子群中产生的。作为说明,我们重现了拉马努金对于$\frac {1}{\pi }$的惊人公式和Cooper等人最近的结果,并为$\frac {1}{\pi }$导出了一个新的有理拉马努金型级数。作为副产品,我们得到了一个一般意义上的clausen型公式,并得到了一个与上述拉马努金公式密切相关的clausen型二次变换公式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
分享
查看原文
Ramanujan type series for , revisited
Abstract In this note, we revisit Ramanujan-type series for $\frac {1}{\pi }$ and show how they arise from genus zero subgroups of $\mathrm {SL}_{2}(\mathbb {R})$ that are commensurable with $\mathrm {SL}_{2}(\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\frac {1}{\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\frac {1}{\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信