{"title":"12 色广义弗罗贝尼斯分区的一些全等式","authors":"Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang","doi":"10.1017/s0013091524000294","DOIUrl":null,"url":null,"abstract":"In his 1984 AMS Memoir, Andrews introduced the family of functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline1.png\"/> <jats:tex-math>$c\\phi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the number of <jats:italic>k</jats:italic>-coloured generalized Frobenius partitions of <jats:italic>n</jats:italic>. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline2.png\"/> <jats:tex-math>$\\textrm{C}\\Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline3.png\"/> <jats:tex-math>$2\\leq k\\leq17$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by utilizing the theory of modular forms, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline4.png\"/> <jats:tex-math>$\\textrm{C}\\Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the generating function of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline5.png\"/> <jats:tex-math>$c\\phi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we first establish another expression of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline6.png\"/> <jats:tex-math>$\\textrm{C}\\Phi_{12}(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> with integer coefficients, then prove some congruences modulo small powers of 3 for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline7.png\"/> <jats:tex-math>$c\\phi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000294_inline8.png\"/> <jats:tex-math>$c\\phi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some congruences for 12-coloured generalized Frobenius partitions\",\"authors\":\"Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang\",\"doi\":\"10.1017/s0013091524000294\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In his 1984 AMS Memoir, Andrews introduced the family of functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline1.png\\\"/> <jats:tex-math>$c\\\\phi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the number of <jats:italic>k</jats:italic>-coloured generalized Frobenius partitions of <jats:italic>n</jats:italic>. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline2.png\\\"/> <jats:tex-math>$\\\\textrm{C}\\\\Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline3.png\\\"/> <jats:tex-math>$2\\\\leq k\\\\leq17$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by utilizing the theory of modular forms, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline4.png\\\"/> <jats:tex-math>$\\\\textrm{C}\\\\Phi_k(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the generating function of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline5.png\\\"/> <jats:tex-math>$c\\\\phi_k(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we first establish another expression of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline6.png\\\"/> <jats:tex-math>$\\\\textrm{C}\\\\Phi_{12}(q)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> with integer coefficients, then prove some congruences modulo small powers of 3 for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline7.png\\\"/> <jats:tex-math>$c\\\\phi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" mimetype=\\\"image\\\" xlink:href=\\\"S0013091524000294_inline8.png\\\"/> <jats:tex-math>$c\\\\phi_{12}(n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/s0013091524000294\",\"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":"100","ListUrlMain":"https://doi.org/10.1017/s0013091524000294","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Some congruences for 12-coloured generalized Frobenius partitions
In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$. In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$.
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