Ramanujan Journal最新文献

{"title":"A positivity conjecture for a quotient of <i>q</i>-binomial coefficients.","authors":"M Gatzweiler, C Krattenthaler","doi":"10.1007/s11139-025-01285-2","DOIUrl":"10.1007/s11139-025-01285-2","url":null,"abstract":"<p><p>We conjecture that, if the quotient of two <i>q</i>-binomial coefficients with the same top argument is a polynomial, then it has non-negative coefficients. We summarise what is known about the conjecture and prove it in two non-trivial cases. Moreover, we move ahead to extend our conjecture to D. Stanton's fake Gaussian sequences. As a corollary of one of our results we obtain that a polynomial that is conjectured to be a cyclic sieving polynomial for Kreweras words [S. Hopkins and M. Rubey, <i>Selecta Math. (N.S.)</i> <b>28</b> (2022), Paper No. 10] is indeed a polynomial with non-negative integer coefficients.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"69 1","pages":"13"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12711936/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145806165","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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Computer-assisted construction of Ramanujan-Sato series for 1 over <ns0:math><ns0:mi>π</ns0:mi></ns0:math>.","authors":"Ralf Hemmecke, Peter Paule, Cristian-Silviu Radu","doi":"10.1007/s11139-026-01352-2","DOIUrl":"10.1007/s11139-026-01352-2","url":null,"abstract":"<p><p>Referring to ideas of Sato and Yang in (Math Z 246:1-19, 2004) described a construction of series for 1 over <math><mi>π</mi></math> starting with a pair (<i>g</i>, <i>h</i>), where <i>g</i> is a modular form of weight 2 and <i>h</i> is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called \"Sato construction\". Series for <math><mrow><mn>1</mn> <mo>/</mo> <mi>π</mi></mrow> </math> obtained this way will be called \"Ramanujan-Sato\" series. Famous series fit into this definition, for instance, Ramanujan's series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of <math><mi>π</mi></math> . We show that these series are induced by members of infinite families of Sato triples <math><mrow><mo>(</mo> <mi>N</mi> <mo>,</mo> <msub><mi>γ</mi> <mi>N</mi></msub> <mo>,</mo> <msub><mi>τ</mi> <mi>N</mi></msub> <mo>)</mo></mrow> </math> where <math><mrow><mi>N</mi> <mo>></mo> <mn>1</mn></mrow> </math> is an integer and <math><msub><mi>γ</mi> <mi>N</mi></msub> </math> a <math><mrow><mn>2</mn> <mo>×</mo> <mn>2</mn></mrow> </math> matrix satisfying <math> <mrow><msub><mi>γ</mi> <mi>N</mi></msub> <msub><mi>τ</mi> <mi>N</mi></msub> <mo>=</mo> <mi>N</mi> <msub><mi>τ</mi> <mi>N</mi></msub> </mrow> </math> for <math><msub><mi>τ</mi> <mi>N</mi></msub> </math> being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm \"ModFormDE\", as described in Paule and Radu in Int J Number Theory (17:713-759, 2021), a central role is played by the algorithm \"MultiSamba\", an extension of Samba (\"subalgebra module basis algorithm\") originating from Radu in (J Symb Comput 68:225-253, 2015) and Hemmecke in (J Symb Comput 84:14-24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for <math><mrow><mn>1</mn> <mo>/</mo> <mi>π</mi></mrow> </math> constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"69 3","pages":"73"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12946297/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147328124","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":"Arithmetic properties of character degrees and the generalised knutson index.","authors":"Diego Martín Duro","doi":"10.1007/s11139-025-01311-3","DOIUrl":"https://doi.org/10.1007/s11139-025-01311-3","url":null,"abstract":"<p><p>In this paper, we introduce the generalised Knutson Index and compute it for the special linear groups and projective special linear groups of degree two by computing the lowest common multiple of the degrees of their irreducible representations. We also classify all alternating and symmetric groups such that the lowest common multiple of the degrees of their irreducible representations equals the order groups, which yields a lower bound on the generalised Knutson Indices of these groups.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"69 2","pages":"37"},"PeriodicalIF":0.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12854403/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146108471","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 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":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Ramanujan's partition generating functions modulo <ns0:math><ns0:mi>ℓ</ns0:mi></ns0:math>.","authors":"Kathrin Bringmann, William Craig, Ken Ono","doi":"10.1007/s11139-025-01241-0","DOIUrl":"https://doi.org/10.1007/s11139-025-01241-0","url":null,"abstract":"&lt;p&gt;&lt;p&gt;For the partition function &lt;i&gt;p&lt;/i&gt;(&lt;i&gt;n&lt;/i&gt;), Ramanujan proved the striking identities &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt; &lt;mn&gt;5&lt;/mn&gt;&lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;5&lt;/mn&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;4&lt;/mn&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt; &lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;5&lt;/mn&gt; &lt;munder&gt;&lt;mo&gt;∏&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mfrac&gt; &lt;msubsup&gt; &lt;mfenced&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt; &lt;/mfenced&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;5&lt;/mn&gt;&lt;/msubsup&gt; &lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;6&lt;/mn&gt;&lt;/msubsup&gt; &lt;/mfrac&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;7&lt;/mn&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;5&lt;/mn&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt; &lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;7&lt;/mn&gt; &lt;munder&gt;&lt;mo&gt;∏&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mfrac&gt; &lt;msubsup&gt; &lt;mfenced&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msup&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msup&gt; &lt;/mfenced&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;3&lt;/mn&gt;&lt;/msubsup&gt; &lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;4&lt;/mn&gt;&lt;/msubsup&gt; &lt;/mfrac&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;49&lt;/mn&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;munder&gt;&lt;mo&gt;∏&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mfrac&gt; &lt;msubsup&gt; &lt;mfenced&gt;&lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msup&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msup&gt; &lt;/mfenced&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;7&lt;/mn&gt;&lt;/msubsup&gt; &lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;mn&gt;8&lt;/mn&gt;&lt;/msubsup&gt; &lt;/mfrac&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; where &lt;math&gt; &lt;mrow&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;;&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mi&gt;∞&lt;/mi&gt;&lt;/msub&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;msub&gt;&lt;mo&gt;∏&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt; &lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;q&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/msup&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;5&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; closed form expressions of the power series &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;P&lt;/mi&gt; &lt;mi&gt;ℓ&lt;/mi&gt;&lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;q&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;munder&gt;&lt;mo&gt;∑&lt;/mo&gt; &lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;≥&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt; &lt;/munder&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;ℓ&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;m","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":"68 3","pages":"85"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12586220/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145460585","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}