{"title":"Log concavity for unimodal sequences.","authors":"Walter Bridges, Kathrin Bringmann","doi":"10.1007/s40993-023-00490-6","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we prove that the number of unimodal sequences of size <i>n</i> is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.</p>","PeriodicalId":43826,"journal":{"name":"Research in Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10730647/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Research in Number Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40993-023-00490-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/12/19 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we prove that the number of unimodal sequences of size n is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.
在本文中，我们证明了大小为 n 的单模态序列的数量是对数凹的。这些是混合假模态的系数，由于第二作者和纳扎罗格鲁最近关于假 Theta 函数的研究，它们具有拉德马赫式精确公式。对于弱全形模形式的（受限）分部和系数，对数凹性和高图兰不等式已经得到了很好的研究，解析证明一般需要精确的渐近级数和误差项。在本文中，我们从单模序列的精确公式出发来进行这一计算。我们希望我们的方法适用于其他混合模拟/虚假模态对象系数的精确公式。
期刊介绍：
Research in Number Theory is an international, peer-reviewed Hybrid Journal covering the scope of the mathematical disciplines of Number Theory and Arithmetic Geometry. The Mission of the Journal is to publish high-quality original articles that make a significant contribution to these research areas. It will also publish shorter research communications (Letters) covering nascent research in some of the burgeoning areas of number theory research. This journal publishes the highest quality papers in all of the traditional areas of number theory research, and it actively seeks to publish seminal papers in the most emerging and interdisciplinary areas here as well. Research in Number Theory also publishes comprehensive reviews.
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