Pub Date : 2024-05-24DOI: 10.1007/s11139-024-00874-x
Neha Elizabeth Thomas, K. Vishnu Namboothiri
The near orthgonality of certain k-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average (frac{1}{k^{s(r+1)}}sum limits _{j=1}^{k^s}j^rc_k^{(s)}(j)) remains positive for all (rge 1). Further, we give a lower bound for (max limits _{N}left| sum limits _{j=1}^{N^s}c_k^{(s)}(j) right| ).
阿尔坎（《数论》140:147-168，2014 年）研究了涉及拉马努扬和的某些 k 向量的近正交性。在此，我们对涉及科恩（Duke Math J 16(2):85-90，1949）定义的拉马努强和的广义化的类似向量进行研究。我们还证明了加权平均数（frac{1}{k^{s(r+1)}}sum limits _{j=1}^{k^s}j^rc_k^{(s)}(j)) 对于所有 (rge 1) 都保持为正。此外，我们给出了 (max limits _{N}left| sum limits _{j=1}^{N^s}c_k^{(s)}(j) right|) 的下限。
{"title":"On near orthogonality of certain k-vectors involving generalized Ramanujan sums","authors":"Neha Elizabeth Thomas, K. Vishnu Namboothiri","doi":"10.1007/s11139-024-00874-x","DOIUrl":"https://doi.org/10.1007/s11139-024-00874-x","url":null,"abstract":"<p>The near orthgonality of certain <i>k</i>-vectors involving the Ramanujan sums were studied by Alkan (J Number Theory 140:147–168, 2014). Here we undertake the study of similar vectors involving a generalization of the Ramanujan sums defined by Cohen (Duke Math J 16(2):85–90, 1949). We also prove that the weighted average <span>(frac{1}{k^{s(r+1)}}sum limits _{j=1}^{k^s}j^rc_k^{(s)}(j))</span> remains positive for all <span>(rge 1)</span>. Further, we give a lower bound for <span>(max limits _{N}left| sum limits _{j=1}^{N^s}c_k^{(s)}(j) right| )</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a generalization of 5-dissections of some infinite q-products","authors":"Channabasavayya, Gedela Kavya Keerthana, Ranganatha Dasappa","doi":"10.1007/s11139-024-00872-z","DOIUrl":"https://doi.org/10.1007/s11139-024-00872-z","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"45 20","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141108710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Exact formula for cubic partitions","authors":"Lukas Mauth","doi":"10.1007/s11139-024-00871-0","DOIUrl":"https://doi.org/10.1007/s11139-024-00871-0","url":null,"abstract":"<p>We obtain an exact formula for the cubic partition function and prove a conjecture by Banerjee, Paule, Radu and Zeng.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1007/s11139-024-00873-y
Nikolay E. Borozenets
In this paper, we build on the recent results of Frank Garvan and Rishabh Sarma as well as classical results of Bruce Berndt in order to establish the 11-dissection of the deviations of the rank and crank modulo 11. Using our new dissections, we re-derive the results of Garvan, Atkin, Swinnerton-Dyer, Hussain, Ekin and Chern. By developing and exploiting positivity conditions for quotients of theta functions, we will also prove new rank–crank inequalities and make several conjectures, one of which was recently solved by Kathrin Bringmann and Badri Vishal Pandey. For other applications of our methods, in this paper, we will also prove new congruences for rank moments as well as the Andrews’ smallest parts function and Eisenstein series.
{"title":"Deviation of the rank and crank modulo 11","authors":"Nikolay E. Borozenets","doi":"10.1007/s11139-024-00873-y","DOIUrl":"https://doi.org/10.1007/s11139-024-00873-y","url":null,"abstract":"<p>In this paper, we build on the recent results of Frank Garvan and Rishabh Sarma as well as classical results of Bruce Berndt in order to establish the 11-dissection of the deviations of the rank and crank modulo 11. Using our new dissections, we re-derive the results of Garvan, Atkin, Swinnerton-Dyer, Hussain, Ekin and Chern. By developing and exploiting positivity conditions for quotients of theta functions, we will also prove new rank–crank inequalities and make several conjectures, one of which was recently solved by Kathrin Bringmann and Badri Vishal Pandey. For other applications of our methods, in this paper, we will also prove new congruences for rank moments as well as the Andrews’ smallest parts function and Eisenstein series.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-18DOI: 10.1007/s11139-024-00841-6
Xinmiao Liu, Jiangxia Hou, Fengxia Liu
Carlitz established a q-analog of the Eulerian numbers (A_{n,k}(q)) and defined the relationship (A_{n,k}(q)=q^{frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q)). In this paper, by using the combinatorial interpretation of (A_{n,k}^{*}(q)) and constructing injective maps, we prove that (A_{n,k}^{*}(q)) and (A_{n,k}(q)) are q-log-concave, that is, all the coefficients of the polynomials (( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) ) and ((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q)) are nonnegative for (1< k <n).
{"title":"A combinatorial proof of q-log-concavity of q-Eulerian numbers","authors":"Xinmiao Liu, Jiangxia Hou, Fengxia Liu","doi":"10.1007/s11139-024-00841-6","DOIUrl":"https://doi.org/10.1007/s11139-024-00841-6","url":null,"abstract":"<p>Carlitz established a <i>q</i>-analog of the Eulerian numbers <span>(A_{n,k}(q))</span> and defined the relationship <span>(A_{n,k}(q)=q^{frac{(n-k)(n-k+1)}{2}}A_{n,k}^{*}(q))</span>. In this paper, by using the combinatorial interpretation of <span>(A_{n,k}^{*}(q))</span> and constructing injective maps, we prove that <span>(A_{n,k}^{*}(q))</span> and <span>(A_{n,k}(q))</span> are <i>q</i>-log-concave, that is, all the coefficients of the polynomials <span>(( A_{n,k}^{*}(q)) ^{2}- A_{n,k-1}^{*}(q) A_{n,k+1}^{*}(q) )</span> and <span>((A_{n,k}(q)) ^{2}- A_{n,k-1}(q) A_{n,k+1}(q))</span> are nonnegative for <span>(1< k <n)</span>.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1007/s11139-024-00860-3
S. Krishnamoorthy, R. Muneeswaran
{"title":"The divisibility of the class number of the imaginary quadratic fields $${mathbb {Q}}(sqrt{1-2m^k})$$","authors":"S. Krishnamoorthy, R. Muneeswaran","doi":"10.1007/s11139-024-00860-3","DOIUrl":"https://doi.org/10.1007/s11139-024-00860-3","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"31 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140968340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1007/s11139-024-00868-9
B. Ramakrishnan, Lalit Vaishya
{"title":"Figurate numbers, forms of mixed type, and their representation numbers","authors":"B. Ramakrishnan, Lalit Vaishya","doi":"10.1007/s11139-024-00868-9","DOIUrl":"https://doi.org/10.1007/s11139-024-00868-9","url":null,"abstract":"","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"48 17","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号