Ramanujan Journal最新文献

{"title":"A remarkable basic hypergeometric identity.","authors":"Christian Krattenthaler, Wadim Zudilin","doi":"10.1007/s11139-024-00994-4","DOIUrl":"https://doi.org/10.1007/s11139-024-00994-4","url":null,"abstract":"<p><p>We give a closed form for <i>quotients</i> of truncated basic hypergeometric series where the base <i>q</i> is evaluated at roots of unity.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"66 3","pages":"48"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11785618/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143081377","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}

{"title":"Flipping operators and locally harmonic Maass forms.","authors":"Kathrin Bringmann, Andreas Mono, Larry Rolen","doi":"10.1007/s11139-025-01183-7","DOIUrl":"https://doi.org/10.1007/s11139-025-01183-7","url":null,"abstract":"<p><p>In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass-Poincaré series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincaré series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincaré series.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"68 2","pages":"40"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12334488/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144818304","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}

{"title":"Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution.","authors":"James A Sellers, Nicolas Allen Smoot","doi":"10.1007/s11139-025-01111-9","DOIUrl":"10.1007/s11139-025-01111-9","url":null,"abstract":"<p><p>For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts <b>cannot</b> be distinct (in the case of POND partitions) or the even parts <b>cannot</b> be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary <i>q</i>-series identities and generating function manipulations, along with mathematical induction: For all <math><mrow><mi>α</mi> <mo>≥</mo> <mn>1</mn></mrow> </math> and all <math><mrow><mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo></mrow> </math> <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow><mrow><mspace></mspace> <mtext>pend</mtext> <mspace></mspace></mrow> <mfenced><msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn></mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac><mrow><mn>17</mn> <mo>·</mo> <msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi></mrow> </msup> <mo>-</mo> <mn>1</mn></mrow> <mn>8</mn></mfrac> </mfenced> </mrow> </mtd> <mtd><mrow><mo>≡</mo> <mn>0</mn> <mspace></mspace> <mo>(</mo> <mo>mod</mo> <mspace></mspace> <mn>3</mn> <mo>)</mo> <mo>,</mo> <mspace></mspace> <mtext>and</mtext></mrow> </mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mrow><mspace></mspace> <mtext>pond</mtext> <mspace></mspace></mrow> <mfenced><msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn></mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac><mrow><mn>23</mn> <mo>·</mo> <msup><mn>3</mn> <mrow><mn>2</mn> <mi>α</mi></mrow> </msup> <mo>+</mo> <mn>1</mn></mrow> <mn>8</mn></mfrac> </mfenced> </mrow> </mtd> <mtd><mrow><mo>≡</mo> <mn>0</mn> <mspace></mspace> <mo>(</mo> <mo>mod</mo> <mspace></mspace> <mn>3</mn> <mo>)</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math> <mrow><mrow><mspace></mspace> <mtext>pend</mtext> <mspace></mspace></mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> counts the number of PEND partitions of weight <i>n</i> and <math> <mrow><mrow><mspace></mspace> <mtext>pond</mtext> <mspace></mspace></mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> counts the number of POND partitions of weight <i>n</i>. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin-Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 3","pages":"60"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12102003/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144144343","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}

{"title":"Kloosterman sums on orthogonal groups.","authors":"Catinca Mujdei","doi":"10.1007/s11139-025-01135-1","DOIUrl":"10.1007/s11139-025-01135-1","url":null,"abstract":"<p><p>We study Kloosterman sums on the orthogonal groups <math><mrow><mi>S</mi> <msub><mi>O</mi> <mrow><mn>3</mn> <mo>,</mo> <mn>3</mn></mrow> </msub> </mrow> </math> and <math><mrow><mi>S</mi> <msub><mi>O</mi> <mrow><mn>4</mn> <mo>,</mo> <mn>2</mn></mrow> </msub> </mrow> </math> , associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and <i>p</i>-adic analysis.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 4","pages":"94"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12185604/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144499341","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}

{"title":"Irrationality and transcendence questions in the 'poor man's adèle ring'.","authors":"Florian Luca, Wadim Zudilin","doi":"10.1007/s11139-025-01132-4","DOIUrl":"https://doi.org/10.1007/s11139-025-01132-4","url":null,"abstract":"<p><p>We discuss arithmetic questions related to the 'poor man's adèle ring' <math><mi>A</mi></math> whose elements are encoded by sequences <math> <msub><mrow><mo>(</mo> <msub><mi>t</mi> <mi>p</mi></msub> <mo>)</mo></mrow> <mi>p</mi></msub> </math> indexed by prime numbers, with each <math><msub><mi>t</mi> <mi>p</mi></msub> </math> viewed as a residue in <math><mrow><mi>Z</mi> <mo>/</mo> <mi>p</mi> <mi>Z</mi></mrow> </math> . Our main theorem is about the <math><mi>A</mi></math> -transcendence of the element <math> <msub><mrow><mo>(</mo> <msub><mi>F</mi> <mi>p</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> <mo>)</mo></mrow> <mi>p</mi></msub> </math> , where <math> <mrow><msub><mi>F</mi> <mi>n</mi></msub> <mrow><mo>(</mo> <mi>q</mi> <mo>)</mo></mrow> </mrow> </math> (Schur's <i>q</i>-Fibonacci numbers) are the (1, 1)-entries of <math><mrow><mn>2</mn> <mo>×</mo> <mn>2</mn></mrow> </math> -matrices <dispformula> <math> <mrow> <mtable> <mtr> <mtd> <mrow> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mn>1</mn></mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <mi>q</mi></mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msup><mi>q</mi> <mn>2</mn></msup> </mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> <mo>⋯</mo> <mfenced> <mrow> <mtable> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>1</mn></mtd> </mtr> <mtr> <mtd><mrow><mrow></mrow> <msup><mi>q</mi> <mrow><mi>n</mi> <mo>-</mo> <mn>2</mn></mrow> </msup> </mrow> </mtd> <mtd><mn>0</mn></mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> and <math><mrow><mi>q</mi> <mo>></mo> <mn>1</mn></mrow> </math> is an integer. This result was previously known for <math><mrow><mi>q</mi> <mo>></mo> <mn>1</mn></mrow> </math> square free under the GRH.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 4","pages":"88"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12177006/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144477899","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}

{"title":"Normal distribution of bad reduction.","authors":"Robert J Lemke Oliver, Daniel Loughran, Ari Shnidman","doi":"10.1007/s11139-025-01108-4","DOIUrl":"https://doi.org/10.1007/s11139-025-01108-4","url":null,"abstract":"<p><p>We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, <math><mrow><mn>100</mn> <mo>%</mo></mrow> </math> of curves in these families have, in a precise sense, many such primes.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"67 3","pages":"52"},"PeriodicalIF":0.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12098207/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144144349","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}

{"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}

{"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}

{"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}

{"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}