Pub Date : 2025-01-01Epub Date: 2025-09-22DOI: 10.1007/s11139-025-01225-0
Runqiao Li, Ali K Uncu
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.
{"title":"A MacMahon analysis view of cylindric partitions.","authors":"Runqiao Li, Ali K Uncu","doi":"10.1007/s11139-025-01225-0","DOIUrl":"10.1007/s11139-025-01225-0","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":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12454573/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145139372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s11139-023-00785-3
Mamta Rani, Avnish K. Sharma, Sharwan K. Tiwari, Anupama Panigrahi
This article studies the existence of elements $$alpha $$ in finite fields $$mathbb {F}_{q^n}$$ such that both $$alpha $$ and its inverse $$alpha ^{-1}$$ are r-primitive and k-normal over $$mathbb {F}_q$$ . We define a characteristic function for the set of k-normal elements and use it to establish a sufficient condition for the existence of the desired pair $$(alpha ,alpha ^{-1})$$ . Moreover, we find that for $$nge 7$$ , there always exists a pair $$(alpha ,alpha ^{-1})$$ of 1-primitive and 1-normal elements in $$mathbb {F}_{q^n}$$ over $$mathbb {F}_q$$ . Additionally, we obtain that for $$n=5,6$$ , if $$textrm{gcd}(q,n)=1$$ , there always exists such a pair in $$mathbb {F}_{q^n}$$ , except for the field $$mathbb {F}_{4^5}$$ .
{"title":"Inverses of r-primitive k-normal elements over finite fields","authors":"Mamta Rani, Avnish K. Sharma, Sharwan K. Tiwari, Anupama Panigrahi","doi":"10.1007/s11139-023-00785-3","DOIUrl":"https://doi.org/10.1007/s11139-023-00785-3","url":null,"abstract":"This article studies the existence of elements $$alpha $$ in finite fields $$mathbb {F}_{q^n}$$ such that both $$alpha $$ and its inverse $$alpha ^{-1}$$ are r-primitive and k-normal over $$mathbb {F}_q$$ . We define a characteristic function for the set of k-normal elements and use it to establish a sufficient condition for the existence of the desired pair $$(alpha ,alpha ^{-1})$$ . Moreover, we find that for $$nge 7$$ , there always exists a pair $$(alpha ,alpha ^{-1})$$ of 1-primitive and 1-normal elements in $$mathbb {F}_{q^n}$$ over $$mathbb {F}_q$$ . Additionally, we obtain that for $$n=5,6$$ , if $$textrm{gcd}(q,n)=1$$ , there always exists such a pair in $$mathbb {F}_{q^n}$$ , except for the field $$mathbb {F}_{4^5}$$ .","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"10 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135266170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s11139-023-00787-1
Bernadette Faye, Florian Luca, Volker Ziegler
Abstract Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S -unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S -unit associated with the set of primes $${2,3,5,7}$$ {2,3,5,7} .
{"title":"On a variant of Pillai’s problem with factorials and S-units","authors":"Bernadette Faye, Florian Luca, Volker Ziegler","doi":"10.1007/s11139-023-00787-1","DOIUrl":"https://doi.org/10.1007/s11139-023-00787-1","url":null,"abstract":"Abstract Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S -unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S -unit associated with the set of primes $${2,3,5,7}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mn>7</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> .","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s11139-023-00786-2
Shaul Zemel
{"title":"Seesaw identities and theta contractions with generalized theta functions, and restrictions of theta lifts","authors":"Shaul Zemel","doi":"10.1007/s11139-023-00786-2","DOIUrl":"https://doi.org/10.1007/s11139-023-00786-2","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"153 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s11139-023-00784-4
Giacomo Cherubini, Pietro Mercuri
{"title":"Parity of the 8-regular partition function","authors":"Giacomo Cherubini, Pietro Mercuri","doi":"10.1007/s11139-023-00784-4","DOIUrl":"https://doi.org/10.1007/s11139-023-00784-4","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"225 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s11139-023-00790-6
Florian Münkel, Lerna Pehlivan, Kenneth S. Williams
{"title":"Infinite product representations of some q-series","authors":"Florian Münkel, Lerna Pehlivan, Kenneth S. Williams","doi":"10.1007/s11139-023-00790-6","DOIUrl":"https://doi.org/10.1007/s11139-023-00790-6","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-27DOI: 10.1007/s11139-023-00744-y
Jane Y. X. Yang, Li Zhou
{"title":"Identities on mex-related partitions","authors":"Jane Y. X. Yang, Li Zhou","doi":"10.1007/s11139-023-00744-y","DOIUrl":"https://doi.org/10.1007/s11139-023-00744-y","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135535696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-26DOI: 10.1007/s11139-023-00782-6
Ce Xu, Jianqiang Zhao
In this paper, we define some weighted sums of the alternating multiple T-values (AMTVs) and study several duality formulas for them. Then we introduce the alternating version of the convoluted T-values and Kaneko–Tsumura $$psi $$ -function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $$mathbb {Q}$$ -vector space generated by the AMTVs of any fixed weight w and provide some evidence for the conjecture that their dimensions $${d_w}_{wge 1}$$ form the tribonacci sequence 1, 2, 4, 7, 13, ....
{"title":"Alternating multiple T-values: weighted sums, duality, and dimension conjecture","authors":"Ce Xu, Jianqiang Zhao","doi":"10.1007/s11139-023-00782-6","DOIUrl":"https://doi.org/10.1007/s11139-023-00782-6","url":null,"abstract":"In this paper, we define some weighted sums of the alternating multiple T-values (AMTVs) and study several duality formulas for them. Then we introduce the alternating version of the convoluted T-values and Kaneko–Tsumura $$psi $$ -function, which are proved to be closely related to the AMTVs. At the end of the paper, we study the $$mathbb {Q}$$ -vector space generated by the AMTVs of any fixed weight w and provide some evidence for the conjecture that their dimensions $${d_w}_{wge 1}$$ form the tribonacci sequence 1, 2, 4, 7, 13, ....","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134904247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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