{"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}
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.