Congruence properties for a certain kind of partition functions

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• 作者：Su-Ping Cui^a ; ^b ; ^{jiayoucui@163.com" class="auth_mail" title="E-mail the corresponding author} ; Nancy S.S. Gu^b ; ^{gu@nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Anthony X. Huang^b ; ^{antho@foxmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11P83 ; 05A17
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：26 February 2016
• 年：2016
• 卷：290
• 期：Complete
• 页码：739-772
• 全文大小：488 K

文摘

In light of the modular equations of fifth and seventh order, we derive some congruence properties for a certain kind of partition functions i1" class="mathmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si1.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=57aa9fe00909465b5c5c8b6d0170b944" title="Click to view the MathML source">a(n)hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">ahy="false">(nhy="false">)h> which satisfy hmlsrc">he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si2.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=01528bf4c6b5a9739123f095c6a667fa">height="18" width="233" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000187081500540X-si2.gif">hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">∑n=0∞ahy="false">(nhy="false">)qn≡hy="false">(q;qhy="false">)∞kh="0.40em">hy="false">(hvariant="normal">modh="0.25em">mhy="false">)h>, where k   is a positive integer with i10" class="mathmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si10.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=98173b686196a92471c77f1dd21837cb" title="Click to view the MathML source">1≤k≤24hContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">1≤k≤24h> and hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si4.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=4b0128c9e5c6c78c47f1173b25c3f4a0" title="Click to view the MathML source">m=2,3hContainer hidden">hCode">h altimg="si4.gif" overflow="scroll">m=2,3h>. In view of these properties, we obtain many infinite families of congruences for hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si5.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=182f8b62e7227ff5fe0071e0b30142d5" title="Click to view the MathML source">cϕ_k(n)hContainer hidden">hCode">h altimg="si5.gif" overflow="scroll">cϕkhy="false">(nhy="false">)h>, the number of generalized Frobenius partitions of n with k   colors, and hmlsrc">he MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000187081500540X&_mathId=si6.gif&_user=111111111&_pii=S000187081500540X&_rdoc=1&_issn=00018708&md5=8e9812318f44e395232d440eb49e5408">height="18" width="48" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000187081500540X-si6.gif">hContainer hidden">hCode">h altimg="si6.gif" overflow="scroll">cϕk‾hy="false">(nhy="false">)h>, the number of generalized Frobenius partitions of n with k colors whose order is k under cyclic permutation of the k colors. Meanwhile, we also apply the main theorems to some other kinds of partition functions.