Arithmetic properties for (s,t)-regular bipartition functions

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• 作者：Ernest X.W. Xia ^{ernestxwxia@163.com" class="auth_mail" title="E-mail the corresponding author} ; Olivia X.M. Yao ; ^{yaoxiangmei@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11P83 ; 05A17
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：171
• 期：Complete
• 页码：1-17
• 全文大小：333 K

文摘

Let af2f79afb3b7a6ffbc3bbce0a" title="Click to view the MathML source">B_s,t(n)$B_{s,t}(n)$ denote the number of (s,t)$(s,t)$-regular bipartitions. Recently, Dou discovered an infinite family of congruences modulo 11 for B_3,11(n)$B_{3,11}(n)$. She also presented several conjectures on af2f79afb3b7a6ffbc3bbce0a" title="Click to view the MathML source">B_s,t(n)$B_{s,t}(n)$. In this paper, utilizing an theta function identity appeared in Berndt's book, we confirm three conjectures on e77416fae" title="Click to view the MathML source">B_3,7(n)$B_{3,7}(n)$ given by Dou. Moreover, we prove several infinite families of congruences modulo 3 and 5 for b952ab31e5b25933ee948" title="Click to view the MathML source">B_3,s(n)$B_{3,s}(n)$ and B_5,s(n)$B_{5,s}(n)$. In addition, we prove many infinite families of congruences modulo 7 for e77416fae" title="Click to view the MathML source">B_3,7(n)$B_{3,7}(n)$ by employing an identity of Newman.