On rational matrix exact covering systems of and its applications to Ramanujan's forty identities

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• 作者：Zhu Cao^a ; ^{zcao@kennesaw.edu" class="auth_mail" title="E-mail the corresponding author} ; Yong Hu^b ; ^{yhu@hust.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Covering systems ; q-series ; Theta functions ; Rogers&ndash ; Ramanujan functions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：322-334
• 全文大小：309 K

文摘

For a nonsingular integer matrix B  , the set of cosets of the quotient module class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304838&_mathId=si17.gif&_user=111111111&_pii=S0022247X16304838&_rdoc=1&_issn=0022247X&md5=34ea6b5e55110ea3a89f0946aed3a323" title="Click to view the MathML source">Zⁿ/BZⁿclass="mathContainer hidden">class="mathCode">${\mathbb{Z}}^n/B{\mathbb{Z}}^n$ forms an exact covering system (ECS) of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304838&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304838&_rdoc=1&_issn=0022247X&md5=8773bbc212922067cab0363518c9ad4e" title="Click to view the MathML source">Zⁿclass="mathContainer hidden">class="mathCode">${\mathbb{Z}}^n$. In this paper, we use the Smith normal form to obtain another type of matrix ECS with rational entries which we call rational matrix ECS. Using rational matrix ECS of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304838&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304838&_rdoc=1&_issn=0022247X&md5=765b7385015435d4a6933fc5fb2ff8f8" title="Click to view the MathML source">Z²class="mathContainer hidden">class="mathCode">${\mathbb{Z}}^2$, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.