{"title":"Asymptotics of solutions of special second-order linear recurrencies with polynomial coefficients and boundary effects of polynomial filters","authors":"Alexey A. Kytmanov , Sergey P. Tsarev","doi":"10.1016/j.jsc.2024.102386","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).</div><div>Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717124000907","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we prove that classical discrete orthogonal polynomials (Hahn polynomials on an equidistant grid with unit weights) of high degrees have extremely small values near the endpoints (we call this property “rapid decay near the endpoints”) but extremely large values between these grid points and their roots are very close to the grid points near the endpoints. These results imply important general boundary effects for stable linear polynomial filters (we call this property “rapid boundary attenuation”).
Our results give interesting examples of nontrivial asymptotics of practically important solutions of special second-order linear recurrencies with polynomial coefficients studied by M.Petkovšek; to his memory we dedicate this paper.
期刊介绍：
An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects.
It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.
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