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 Zⁿ/BZⁿ${\mathbb{Z}}^n/B{\mathbb{Z}}^n$ forms an exact covering system (ECS) of cab0363518c9ad4e" title="Click to view the MathML source">Zⁿ${\mathbb{Z}}^n$. 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 Z²${\mathbb{Z}}^2$, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.