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.
| Original language | English |
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| Article number | 71 |
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| Journal | The Ramanujan Journal |
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| Volume | 68 |
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| Early online date | 22 Sept 2025 |
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| DOIs | |
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| Publication status | Published - 22 Sept 2025 |
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The second author acknowledges the support of the Austrian Science Fund FWF, P34501N, and UKRI EPSRC EP/T015713/1 projects.Mathematica notebooks used to certify the results of this paper are shared as supplementary files on the journal servers, on ArXiv, and on the second author’s website openly. No datasets were generated or analysed during the current study.
The authors express gratitude to George E. Andrews, Alexander Berkovich, Kağan Kurşungöz, and Ae Ja Yee for their encouragement and comments on the manuscript. The authors would also like to thank Ole Warnaar for his thoughtful comments and corrections. We thank Feihu Liu for pointing out the k -elongated partition diamond connection. We thank the anonymous referees for their careful reading, comments, and questions.
