{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T13:13:47Z","timestamp":1753881227350,"version":"3.41.2"},"reference-count":19,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2012,9,25]],"date-time":"2012-09-25T00:00:00Z","timestamp":1348531200000},"content-version":"vor","delay-in-days":268,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11JDG035","11JDG036"],"award-info":[{"award-number":["11JDG035","11JDG036"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002703","name":"Jiangsu University","doi-asserted-by":"publisher","award":["11JDG035","11JDG036"],"award-info":[{"award-number":["11JDG035","11JDG036"]}],"id":[{"id":"10.13039\/501100002703","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002703","name":"Jiangsu University","doi-asserted-by":"publisher","award":["11JDG035","11JDG036"],"award-info":[{"award-number":["11JDG035","11JDG036"]}],"id":[{"id":"10.13039\/501100002703","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>Based on the theories of Ramanujan\u2032s elliptic functions and the (<jats:italic>p<\/jats:italic>, <jats:italic>k<\/jats:italic>)\u2010parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins\u2032 cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins\u2032 cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.<\/jats:p>","DOI":"10.1155\/2012\/181264","type":"journal-article","created":{"date-parts":[[2012,9,26]],"date-time":"2012-09-26T07:40:42Z","timestamp":1348645242000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Eisenstein Series Identities Involving the Borweins\u2032 Cubic Theta Functions"],"prefix":"10.1155","volume":"2012","author":[{"given":"Ernest X. W.","family":"Xia","sequence":"first","affiliation":[]},{"given":"Olivia X. M.","family":"Yao","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,9,25]]},"reference":[{"key":"e_1_2_6_1_2","doi-asserted-by":"publisher","DOI":"10.2307\/2001551"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1090\/stml\/034"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.2307\/2154520"},{"volume-title":"Notebooks, Vols. 1, 2","year":"1957","author":"Ramanujan S.","key":"e_1_2_6_4_2"},{"key":"e_1_2_6_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/2155035"},{"key":"e_1_2_6_6_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004198002643"},{"key":"e_1_2_6_7_2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1009825922114"},{"key":"e_1_2_6_8_2","doi-asserted-by":"publisher","DOI":"10.1006\/jnth.2000.2543"},{"key":"e_1_2_6_9_2","doi-asserted-by":"publisher","DOI":"10.4064\/aa124-2-4"},{"key":"e_1_2_6_10_2","doi-asserted-by":"publisher","DOI":"10.12988\/imf.2007.07003"},{"key":"e_1_2_6_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jnt.2006.10.004"},{"key":"e_1_2_6_12_2","first-page":"27","article-title":"Evaluation of the convolution sums \u2211\n                     l+12m=n\n                  \n                  \u03c3(l)\u03c3(m) and \u22113l+4m=n\n                  \n                  \u03c3(l)\u03c3(m)","volume":"1","author":"Alaca A.","year":"2006","journal-title":"Advances in Theoretical and Applied Mathematics"},{"key":"e_1_2_6_13_2","first-page":"93","article-title":"Evaluation of the convolution sums \u2211\n                     l+24m=n\n                  \n                  \u03c3(l)\u03c3(m) and \u22113l+8m=n\n                  \n                  \u03c3(l)\u03c3(m)","volume":"49","author":"Alaca A.","year":"2007","journal-title":"Mathematical Journal of Okayama University"},{"key":"e_1_2_6_14_2","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-2008-001-1"},{"key":"e_1_2_6_15_2","first-page":"229","volume-title":"Number Theory for the Millennium, II","author":"Huard J. 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