Ramanujan Journal最新文献

{"title":"Log-convexity and the overpartition function.","authors":"Gargi Mukherjee","doi":"10.1007/s11139-022-00578-0","DOIUrl":"https://doi.org/10.1007/s11139-022-00578-0","url":null,"abstract":"<p><p>Let <math> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </math> denote the overpartition function. In this paper, we obtain an inequality for the sequence <math> <mrow><msup><mi>Δ</mi> <mn>2</mn></msup> <mo>log</mo> <mspace></mspace> <mroot> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mo>/</mo> <msup><mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mi>α</mi></msup> </mrow> <mrow><mi>n</mi> <mo>-</mo> <mn>1</mn></mrow> </mroot> </mrow> </math> which states that <dispformula> <math> <mrow> <mtable><mtr><mtd></mtd> <mtd><mrow><mo>log</mo> <mrow><mo>(</mo></mrow> <mn>1</mn> <mo>+</mo> <mfrac><mrow><mn>3</mn> <mi>π</mi></mrow> <mrow><mn>4</mn> <msup><mi>n</mi> <mrow><mn>5</mn> <mo>/</mo> <mn>2</mn></mrow> </msup> </mrow> </mfrac> <mo>-</mo> <mfrac><mrow><mn>11</mn> <mo>+</mo> <mn>5</mn> <mi>α</mi></mrow> <msup><mi>n</mi> <mrow><mn>11</mn> <mo>/</mo> <mn>4</mn></mrow> </msup> </mfrac> <mrow><mo>)</mo></mrow> <mo><</mo> <msup><mi>Δ</mi> <mn>2</mn></msup> <mo>log</mo> <mspace></mspace> <mroot> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mo>/</mo> <msup><mrow><mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mi>α</mi></msup> </mrow> <mrow><mi>n</mi> <mo>-</mo> <mn>1</mn></mrow> </mroot> </mrow> </mtd> </mtr> <mtr><mtd><mrow></mrow></mtd> <mtd><mrow><mo><</mo> <mo>log</mo> <mrow><mo>(</mo></mrow> <mn>1</mn> <mo>+</mo> <mfrac><mrow><mn>3</mn> <mi>π</mi></mrow> <mrow><mn>4</mn> <msup><mi>n</mi> <mrow><mn>5</mn> <mo>/</mo> <mn>2</mn></mrow> </msup> </mrow> </mfrac> <mrow><mo>)</mo></mrow> <mspace></mspace> <mspace></mspace> <mtext>for</mtext> <mspace></mspace> <mi>n</mi> <mo>≥</mo> <mi>N</mi> <mrow><mo>(</mo> <mi>α</mi> <mo>)</mo></mrow> <mo>,</mo></mrow> </mtd> </mtr> </mtable> </mrow> </math> </dispformula> where <math><mi>α</mi></math> is a non-negative real number, <math><mrow><mi>N</mi> <mo>(</mo> <mi>α</mi> <mo>)</mo></mrow> </math> is a positive integer depending on <math><mi>α</mi></math> , and <math><mi>Δ</mi></math> is the difference operator with respect to <i>n</i>. This inequality consequently implies <math><mo>log</mo></math> -convexity of <math> <mrow><mrow><mo>{</mo></mrow> <mroot> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> <mo>/</mo> <mi>n</mi></mrow> <mi>n</mi></mroot> <msub><mrow><mo>}</mo></mrow> <mrow><mi>n</mi> <mo>≥</mo> <mn>19</mn></mrow> </msub> </mrow> </math> and <math> <mrow><mrow><mo>{</mo></mrow> <mroot> <mrow><mover><mi>p</mi> <mo>¯</mo></mover> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> <mi>n</mi></mroot> <msub><mrow><mo>}</mo></mrow> <mrow><mi>n</mi> <mo>≥</mo> <mn>4</mn></mrow> </msub> </mrow> </math> . Moreover, it also establishes the asymptotic growth of <math> <mrow><msup><mi>Δ</mi> <mn>2</mn></msup> <mo>log</mo> <mspace></mspace> <mroot> <mro","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9883361/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10601993","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 single-variable proof of the omega SPT congruence family over powers of 5.","authors":"Nicolas Allen Smoot","doi":"10.1007/s11139-023-00747-9","DOIUrl":"10.1007/s11139-023-00747-9","url":null,"abstract":"<p><p>In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function <math><mrow><mi>ω</mi><mo>(</mo><mi>q</mi><mo>)</mo></mrow></math>. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 <math><mrow><mi>Z</mi><mo>[</mo><mi>X</mi><mo>]</mo></mrow></math>-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of <math><mrow><mi>Z</mi><mo>[</mo><mi>X</mi><mo>]</mo></mrow></math>. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10447328/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10164530","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":"On the infinite Borwein product raised to a positive real power.","authors":"Michael J Schlosser,&nbsp;Nian Hong Zhou","doi":"10.1007/s11139-021-00519-3","DOIUrl":"10.1007/s11139-021-00519-3","url":null,"abstract":"<p><p>In this paper, we study properties of the coefficients appearing in the <i>q</i>-series expansion of <math><mrow><msub><mo>∏</mo><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><msup><mrow><mo>[</mo><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>q</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>q</mi><mrow><mi>pn</mi></mrow></msup><mo>)</mo></mrow><mo>]</mo></mrow><mi>δ</mi></msup></mrow></math>, the infinite Borwein product for an arbitrary prime <i>p</i>, raised to an arbitrary positive real power <math><mi>δ</mi></math>. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients. For <math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math> we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent <math><mi>δ</mi></math> is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the <math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math> case.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10185621/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9544955","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":"Sequences in overpartitions.","authors":"George E Andrews,&nbsp;Ali K Uncu","doi":"10.1007/s11139-022-00685-y","DOIUrl":"10.1007/s11139-022-00685-y","url":null,"abstract":"<p><p>This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10185600/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9501289","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":"On signatures of elliptic curves and modular forms","authors":"A. Dąbrowski, J. Pomykala, Sudhir Pujahari","doi":"10.1007/s11139-022-00678-x","DOIUrl":"https://doi.org/10.1007/s11139-022-00678-x","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47977954","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":"Diagonalizable Thue equations: revisited","authors":"N. Saradha, Divyum Sharma","doi":"10.1007/s11139-022-00682-1","DOIUrl":"https://doi.org/10.1007/s11139-022-00682-1","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41699070","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":"Proof of a conjecture of Sun and its extension by Guo","authors":"Wei Xia","doi":"10.1007/s11139-022-00668-z","DOIUrl":"https://doi.org/10.1007/s11139-022-00668-z","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46747458","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":"Sums of k-th powers and Fourier coefficients of cusp forms","authors":"Zhining Wei","doi":"10.1007/s11139-022-00677-y","DOIUrl":"https://doi.org/10.1007/s11139-022-00677-y","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45503834","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":"Expansions over Legendre polynomials and infinite double series identities","authors":"W. Chu, J. Campbell","doi":"10.1007/s11139-022-00663-4","DOIUrl":"https://doi.org/10.1007/s11139-022-00663-4","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49575337","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":"An asymptotic expansion of the hyberbolic umbilic catastrophe integral","authors":"Chelo Ferreira, J. López, Ester Pérez Sinusía","doi":"10.1007/s11139-022-00675-0","DOIUrl":"https://doi.org/10.1007/s11139-022-00675-0","url":null,"abstract":"","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52861085","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}