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 xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=d50fb43af2f79afb3b7a6ffbc3bbce0a" title="Click to view the MathML source">B_s,t(n)$B_{s,t}(n)$ denote the number of xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=2f1ac20d87502155f371f6d7d199ec35" title="Click to view the MathML source">(s,t)$(s,t)$-regular bipartitions. Recently, Dou discovered an infinite family of congruences modulo 11 for xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=9c3a02edb34d7318437375e400b32c2f" title="Click to view the MathML source">B_3,11(n)$B_{3,11}(n)$. She also presented several conjectures on xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=d50fb43af2f79afb3b7a6ffbc3bbce0a" 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 xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=a618ce4792d6717955ec3f7e77416fae" 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 xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=fa721015daeb952ab31e5b25933ee948" title="Click to view the MathML source">B_3,s(n)$B_{3,s}(n)$ and xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=267fa37b822c0e0114dd2b53313227fb" title="Click to view the MathML source">B_5,s(n)$B_{5,s}(n)$. In addition, we prove many infinite families of congruences modulo 7 for xt stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301688&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301688&_rdoc=1&_issn=0022314X&md5=a618ce4792d6717955ec3f7e77416fae" title="Click to view the MathML source">B_3,7(n)$B_{3,7}(n)$ by employing an identity of Newman.