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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
Pub Date : 2023-09-15DOI: 10.1007/s11139-023-00780-8
Robert Schneider, Andrew V. Sills
{"title":"Composition-theoretic series in partition theory","authors":"Robert Schneider, Andrew V. Sills","doi":"10.1007/s11139-023-00780-8","DOIUrl":"https://doi.org/10.1007/s11139-023-00780-8","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135437152","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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