{"title":"从沃森 3 F 2 求和的 q 类比中得出的一些 q-supercongruences","authors":"Victor J. W. Guo","doi":"10.1515/forum-2023-0475","DOIUrl":null,"url":null,"abstract":"We give some <jats:italic>q</jats:italic>-supercongruences from a <jats:italic>q</jats:italic>-analogue of Watson’s <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>2</m:mn> <m:none /> <m:mprescripts /> <m:mn>3</m:mn> <m:none /> </m:mmultiscripts> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0206.png\" /> <jats:tex-math>{{}_{3}F_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> summation and the method of “creative microscoping”, introduced by the author and Zudilin. These <jats:italic>q</jats:italic>-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0181.png\" /> <jats:tex-math>{p^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0475_eq_0180.png\" /> <jats:tex-math>{p^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for any odd prime <jats:italic>p</jats:italic>. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its <jats:italic>q</jats:italic>-analogue.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"233 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation\",\"authors\":\"Victor J. W. Guo\",\"doi\":\"10.1515/forum-2023-0475\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We give some <jats:italic>q</jats:italic>-supercongruences from a <jats:italic>q</jats:italic>-analogue of Watson’s <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mmultiscripts> <m:mi>F</m:mi> <m:mn>2</m:mn> <m:none /> <m:mprescripts /> <m:mn>3</m:mn> <m:none /> </m:mmultiscripts> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0475_eq_0206.png\\\" /> <jats:tex-math>{{}_{3}F_{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> summation and the method of “creative microscoping”, introduced by the author and Zudilin. These <jats:italic>q</jats:italic>-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>p</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0475_eq_0181.png\\\" /> <jats:tex-math>{p^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0475_eq_0180.png\\\" /> <jats:tex-math>{p^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for any odd prime <jats:italic>p</jats:italic>. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its <jats:italic>q</jats:italic>-analogue.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":\"233 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-03-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2023-0475\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0475","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们从沃森的 F 2 3 {{}_{3}F_{2}} 求和的 q-analogue 以及作者和祖迪林提出的 "创造性微观 "方法中给出了一些 q-supercongruences 。这些 q 超共形可以看作是凡-哈姆（Van Hamme）对任意奇素数 p 的 p 3 {p^{3}} 或 p 2 {p^{2}} 模的 (A.2) 超共形的进一步推广。
Some q-supercongruences from a q-analogue of Watson's 3 F 2 summation
We give some q-supercongruences from a q-analogue of Watson’s F23{{}_{3}F_{2}} summation and the method of “creative microscoping”, introduced by the author and Zudilin. These q-supercongruences may be considered as further generalizations of the (A.2) supercongruence of Van Hamme modulo p3{p^{3}} or p2{p^{2}} for any odd prime p. Meanwhile, we confirm a supercongruence conjecture of Wang and Yue through establishing its q-analogue.
期刊介绍：
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.
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