Recent development of special functions from the viewpoint of the representation theory and the integrals in complex variables

2015 Fiscal Year Final Research Report

• Project/Area Number: 23340002
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University (2014-2015); Tokyo Institute of Technology (2011-2013)
• Principal Investigator: MIMACHI KATSUHISA 大阪大学, 情報科学研究科, 教授 (40211594)
• Co-Investigator(Renkei-kenkyūsha): YOSHIDA MASAAKI 九州大学, 名誉教授 (30030787) ; KUROKAWA NOBUSHIGE 東京工業大学, 大学院理工学研究科, 教授 (70114866) ; TAKATA TOSHIE 九州大学, 大学院数理学研究院, 准教授 (40253398)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: 超幾何函数 / 超幾何積分 / 複素積分 / モノドロミー / 既約性 / ねじれホモロジー / フックス型微分方程式 / 接続問題

Outline of Final Research Achievements

We calculated explicitly the circuit matrices associated with Gauss' {}_2F1, the generalized hypergeometric functions {}_{n+1}F_n, Apell's F_1, F_2, F_3, Jordan-Pochhammer F_{JP}, and Lauricella's F_D, and, by using them, we determine the conditionｓ that the corresponding (system of) diffferential equations being irreducible. On the other hand, in the cases of F_2, F_3, F_4, we study the contiguity relations and give the conditionｓ that the corresponding systemｓ being reducible.

We prove that, in each case of the classical hypergeometric equations and Gelfand's hypergeometric system on the space of point configurations, the integral of a multivalued functions over any cycle satisfies the system of differential equations. Here the classical hypergeometric equations mean the equations satisfied by Appell's F_1, F_2, F_3, F_4, Lauricella's F_D, F_A, F_B, F_C, and the generalized hypergeometric function {}_{n+1}F_n.

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