{"title":"Ferrers Functions of Arbitrary Degree and Order and Related Functions","authors":"","doi":"10.1007/s00365-024-09683-3","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P<span> <span>\\(_{\\nu }^{-\\mu }\\left( \\tanh \\left( \\alpha +\\beta \\right) \\right) \\)</span> </span> are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.</p>","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":"30 1","pages":""},"PeriodicalIF":2.3000,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Constructive Approximation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00365-024-09683-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Simple proofs of four generating functions for Ferrers functions are given. Addition theorems for P\(_{\nu }^{-\mu }\left( \tanh \left( \alpha +\beta \right) \right) \) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.
期刊介绍：
Constructive Approximation is an international mathematics journal dedicated to Approximations and Expansions and related research in computation, function theory, functional analysis, interpolation spaces and interpolation of operators, numerical analysis, space of functions, special functions, and applications.
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