{"title":"On the linear independence of -adic polygamma values","authors":"Makoto Kawashima, Anthony Poëls","doi":"10.1112/mtk.70040","DOIUrl":null,"url":null,"abstract":"<p>In this article, we present a new linear independence criterion for values of the <span></span><math></math>-adic polygamma functions defined by Diamond. As an application, we obtain the linear independence of some families of values of the <span></span><math></math>-adic Hurwitz zeta function <span></span><math></math> at distinct shifts <span></span><math></math>. This improves and extends a previous result due to Bel (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) IX (2010), 189–227), as well as irrationality results established by Beukers (Acta Math. Sin. 24 (2008), 663–686). Our proof is based on a novel and explicit construction of Padé-type approximants of the second kind of Diamond's <span></span><math></math>-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"71 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/mtk.70040","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we present a new linear independence criterion for values of the -adic polygamma functions defined by Diamond. As an application, we obtain the linear independence of some families of values of the -adic Hurwitz zeta function at distinct shifts . This improves and extends a previous result due to Bel (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) IX (2010), 189–227), as well as irrationality results established by Beukers (Acta Math. Sin. 24 (2008), 663–686). Our proof is based on a novel and explicit construction of Padé-type approximants of the second kind of Diamond's -adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
