文摘
For a nonsingular integer matrix B , the set of cosets of the quotient module pan id="mmlsi17" class="mathmlsrc">pan 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">Zp>np>/BZp>np>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> forms an exact covering system (ECS) of pan id="mmlsi1" class="mathmlsrc">pan 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">Zp>np>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>. 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 pan id="mmlsi3" class="mathmlsrc">pan 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">Zp>2p>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>, we prove eight identities in Ramanujan's list of forty identities for the Rogers–Ramanujan functions, as well as some other identities.