{"title":"The MacMahon q-Catalan is Convex","authors":"Tewodros Amdeberhan, Stephan Wagner","doi":"10.1007/s00026-023-00677-9","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(n\\ge 2\\)</span> be an integer. We prove the convexity of the so-called MacMahon <i>q</i>-Catalan polynomials <span>\\(C_n(q)=\\frac{1}{[n+1]_q}\\left[ 2n \\atop n \\right] _q\\)</span> viewed as functions of <i>q</i> over the entire set of reals. Along the way, several auxiliary properties of the <i>q</i>-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00677-9.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00677-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let \(n\ge 2\) be an integer. We prove the convexity of the so-called MacMahon q-Catalan polynomials \(C_n(q)=\frac{1}{[n+1]_q}\left[ 2n \atop n \right] _q\) viewed as functions of q over the entire set of reals. Along the way, several auxiliary properties of the q-Catalan polynomials and intermediate results in the form of inequalities are presented, with the aim to make the paper self-contained. We also include a commentary on the convexity of the generating function for the integer partitions.
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