{"title":"与appeell的f2、horn的h2和olsson的fp函数相关的连接公式","authors":"K. Mimachi","doi":"10.2206/kyushujm.74.15","DOIUrl":null,"url":null,"abstract":"Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2) [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2\\ { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"CONNECTION FORMULAS RELATED WITH APPELL'S F2, HORN'S H2 AND OLSSON'S FP FUNCTIONS\",\"authors\":\"K. Mimachi\",\"doi\":\"10.2206/kyushujm.74.15\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2) [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2\\\\ { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.\",\"PeriodicalId\":49929,\"journal\":{\"name\":\"Kyushu Journal of Mathematics\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2020-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Kyushu Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2206/kyushujm.74.15\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Kyushu Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2206/kyushujm.74.15","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
摘要
本文用欧拉型积分法解决了由apappell的F2函数满足的微分方程组E2的一些连接问题。本文的结果再次证明了由Olsson. 0得到的与Appell 's F2、Horn 's H2和Olsson 's FP函数相关的连接公式。Appell的超几何函数F2是F2(a, b1, b2, c1, c2;x, y) =∑m, n≥0 m (a) + n (b1) m (b2) n m ! n ! (c1) m (c2) n xm yn, x y | + | | | < 1, (a) n = 0 (a + n) / 0 (a),并满足系统E2的排名四(AKdF,呃):(E2)[x(1−x)∂2∂x2−xy∂2∂x∂y + {c1−(+ b1 + 1) x}∂∂x−b1 y∂∂y−有所)F = 0, [y(1−y)∂2∂y2−xy∂2∂x∂y + {c2−(+ b2 + 1) y}∂∂y−b2x∂∂x−ab2) F = 0,这是空间上定义c2 \ {{x = 0}∪{x = 1}∪{y = 0}∪{y = 1}∪{x + y = 1}}⊂(P1) 2。在[Ol]中,Olsson证明了当|x/(1−y)|< 1或(0,∞)时E2绕点(0,1)的基本解集由Horn的超几何函数H2和Olsson的超几何函数FP给出,而绕点(0,0)的基本解集由F2给出。并推导出F2、H2与2010数学学科分类相关的关联公式:Primary 33C60;次级33C65, 33C70。
CONNECTION FORMULAS RELATED WITH APPELL'S F2, HORN'S H2 AND OLSSON'S FP FUNCTIONS
Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell’s F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell’s F2, Horn’s H2 and Olsson’s FP functions, which are obtained by Olsson. 0. Introduction Appell’s hypergeometric function F2 is the analytic continuation of F2(a, b1, b2, c1, c2; x, y)= ∑ m,n≥0 (a)m+n(b1)m(b2)n m!n!(c1)m(c2)n xm yn, |x | + |y|< 1, where (a)n = 0(a + n)/0(a), and satisfies the system E2 of rank four [AKdF, Er]: (E2) [ x(1− x) ∂2 ∂x2 − xy ∂2 ∂x∂y + {c1 − (a + b1 + 1)x} ∂ ∂x − b1 y ∂ ∂y − ab1 ] F = 0, [ y(1− y) ∂2 ∂y2 − xy ∂2 ∂x∂y + {c2 − (a + b2 + 1)y} ∂ ∂y − b2x ∂ ∂x − ab2 ] F = 0, which is defined on the space C2\ { {x = 0} ∪ {x = 1} ∪ {y = 0} ∪ {y = 1} ∪ {x + y = 1} } ⊂ (P1)2. In [Ol], Olsson shows that a fundamental set of solutions of E2 around the point (0, 1) in the case |x/(1− y)|< 1 or that around the point (0,∞) is given by Horn’s hypergeometric function H2 and Olsson’s hypergeometric function FP , while that around the point (0, 0) is given by F2. Moreover, he also derives some connection formulas related with F2, H2 and 2010 Mathematics Subject Classification: Primary 33C60; Secondary 33C65, 33C70.
期刊介绍:
The Kyushu Journal of Mathematics is an academic journal in mathematics, published by the Faculty of Mathematics at Kyushu University since 1941. It publishes selected research papers in pure and applied mathematics. One volume, published each year, consists of two issues, approximately 20 articles and 400 pages in total.
More than 500 copies of the journal are distributed through exchange contracts between mathematical journals, and available at many universities, institutes and libraries around the world. The on-line version of the journal is published at "Jstage" (an aggregator for e-journals), where all the articles published by the journal since 1995 are accessible freely through the Internet.