{"title":"A MacMahon analysis view of cylindric partitions.","authors":"Runqiao Li, Ali K Uncu","doi":"10.1007/s11139-025-01225-0","DOIUrl":null,"url":null,"abstract":"<p><p>We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some <i>q</i>-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companion to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.</p><p><strong>Supplementary information: </strong>The online version contains supplementary material available at 10.1007/s11139-025-01225-0.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"68 3","pages":"71"},"PeriodicalIF":0.7000,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12454573/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ramanujan Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11139-025-01225-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/9/22 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companion to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.
Supplementary information: The online version contains supplementary material available at 10.1007/s11139-025-01225-0.
期刊介绍：
The Ramanujan Journal publishes original papers of the highest quality in all areas of mathematics influenced by Srinivasa Ramanujan. His remarkable discoveries have made a great impact on several branches of mathematics, revealing deep and fundamental connections.
The following prioritized listing of topics of interest to the journal is not intended to be exclusive but to demonstrate the editorial policy of attracting papers which represent a broad range of interest:
Hyper-geometric and basic hyper-geometric series (q-series) * Partitions, compositions and combinatory analysis * Circle method and asymptotic formulae * Mock theta functions * Elliptic and theta functions * Modular forms and automorphic functions * Special functions and definite integrals * Continued fractions * Diophantine analysis including irrationality and transcendence * Number theory * Fourier analysis with applications to number theory * Connections between Lie algebras and q-series.
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