{"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}
引用次数: 0
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.