{"title":"广义超几何函数p+1Fp的矩阵代数无穷积表示","authors":"Metin Demi̇ralp, Sevda Üsküplü","doi":"10.1002/anac.200410036","DOIUrl":null,"url":null,"abstract":"<p>We present a novel representation for generalized hypergeometric functions of type <sub><i>p</i>+1</sub><i>F<sub>p</sub></i> which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type <sub><i>p</i>+1</sub><i>F<sub>p</sub></i>. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)</p>","PeriodicalId":100108,"journal":{"name":"Applied Numerical Analysis & Computational Mathematics","volume":"2 2","pages":"212-226"},"PeriodicalIF":0.0000,"publicationDate":"2005-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/anac.200410036","citationCount":"3","resultStr":"{\"title\":\"Matrix Algebraic Infinite Product Representation for Generalized Hypergeometric Functions of Type p+1Fp\",\"authors\":\"Metin Demi̇ralp, Sevda Üsküplü\",\"doi\":\"10.1002/anac.200410036\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present a novel representation for generalized hypergeometric functions of type <sub><i>p</i>+1</sub><i>F<sub>p</sub></i> which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type <sub><i>p</i>+1</sub><i>F<sub>p</sub></i>. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)</p>\",\"PeriodicalId\":100108,\"journal\":{\"name\":\"Applied Numerical Analysis & Computational Mathematics\",\"volume\":\"2 2\",\"pages\":\"212-226\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2005-07-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1002/anac.200410036\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Numerical Analysis & Computational Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/anac.200410036\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Numerical Analysis & Computational Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/anac.200410036","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3