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Algebraic analysis of parametric Stokes phenomena

Research Project

Project/Area Number 26400126
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionKindai University

Principal Investigator

AOKI TAKASHI  近畿大学, 理工学部, 教授 (80159285)

Co-Investigator(Kenkyū-buntansha) 中村 弥生  近畿大学, 理工学部, 准教授 (60388494)
鈴木 貴雄  近畿大学, 理工学部, 准教授 (60527208)
Co-Investigator(Renkei-kenkyūsha) HONDA Naofumi  北海道大学, 大学院・理学系研究院, 准教授 (00238817)
KAWAI Takahiro  京都大学, 数理解析研究所, 名誉教授 (20027379)
TAKEI Yoshitsugu  京都大学, 数理解析研究所, 准教授 (00212019)
YAMAZAKI Susumu  日本大学, 理工学部, 教授 (00349953)
KOIKE Tatsuya  神戸大学, 大学院・理学研究科, 准教授 (80324599)
UMETA Yoko  城西大学, 理学部, 准教授 (90606386)
Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords超幾何微分方程式 / 超幾何関数 / WKB解 / 特異摂動 / ストークス現象 / 漸近展開 / ボレル総和法 / 合流型超幾何微分方程式 / Stokes曲線 / Stokes現象 / Borel総和法 / 無限階微分作用素 / Watsonの補題 / Voros係数 / Borel和
Outline of Final Research Achievements

Introducing a large parameter in the 3 parameters contained in the Gauss hypergeometric differential equation, we can construct the WKB solutions which are formal solutions to the equation. The construction is done algebraically and elementarily, however, these formal solutions are divergent in general and do not have analytic sense. We may apply the Borel resummation method to the formal solutions and can construct analytic solutions and bases of the solution space. On the other hand, the Gauss hypergeometric differential equation has standard bases of solutions expressed by the hypergeometric function. In this research, we have obtained linear relations between these two classes of bases. As an application, asymptotic expansion formulas with respect to the large parameter of the Gauss hypergeometric function have been obtained. At the same time, we have some formulas which describe the parametric Stokes phenomena of the WKB solutions.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (35 results)

All 2018 2017 2016 2015 2014 Other

All Int'l Joint Research (2 results) Journal Article (10 results) (of which Peer Reviewed: 10 results,  Open Access: 5 results,  Acknowledgement Compliant: 4 results) Presentation (21 results) (of which Int'l Joint Research: 9 results,  Invited: 16 results) Remarks (2 results)

  • [Int'l Joint Research] Chapman University(米国)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] Politecnico di Milano(イタリア)

    • Related Report
      2016 Research-status Report
  • [Journal Article] A Higher Order Painlev&eacute; System in Two Variables and Extensions of the Appell Hypergeometric Functions <i>F</i><sub>1</sub>, <i>F</i><sub>2</sub> and <i>F</i><sub>3</sub>2018

    • Author(s)
      Takao Suzuki
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: 61 Issue: 1 Pages: 81-107

    • DOI

      10.1619/fesi.61.81

    • NAID

      130006726045

    • ISSN
      0532-8721
    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Relation between the hypergeometric function and WKB solutions2017

    • Author(s)
      T. Aoki, T. Takahasi and M. Tanda
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: 61

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Foundation of symbol theory for analytic pseudodifferential operators, I2017

    • Author(s)
      Takashi Aoki, Naofumi Honda and Susumu Yamazaki
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 69 Issue: 4 Pages: 1715-1801

    • DOI

      10.2969/jmsj/06941715

    • NAID

      130006887116

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A reformulation of the generalized $q$-Painlev\'{e} VI system with $W(A^{(1)}_{2n+1})$ symmetry2017

    • Author(s)
      Takao Suzuki
    • Journal Title

      J. Integrable Syst.

      Volume: 2 Issue: 1 Pages: 1-18

    • DOI

      10.1093/integr/xyw017

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Parametric Stokes phenomena of the Gauss hypergeometric differential equation with a large parameter2016

    • Author(s)
      Takashi Aoki and Mika Tanda
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 68 Issue: 3 Pages: 1099-1132

    • DOI

      10.2969/jmsj/06831099

    • NAID

      130005166137

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The hypergeometric function and WKB solutions2016

    • Author(s)
      Takashi Aoki, Toshinori Takahashi, Mika Tanda
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: 57 Pages: 61-68

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The b-function of Reiffen's (p,4) isolated singularity2016

    • Author(s)
      Yayoi Nakamura
    • Journal Title

      Journal of algebra and its applications

      Volume: 15 Issue: 06 Pages: 1650115-1650115

    • DOI

      10.1142/s0219498816501152

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] A q-analogue of the Drinfeld-Sokolov hierarchy of type A and q-Painleve; system2015

    • Author(s)
      Takao Suzuki
    • Journal Title

      AMS Contemp. Math.

      Volume: 651 Pages: 25-38

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Exact WKB analysis of confluent hypergeometric differential equations with a large parameter2014

    • Author(s)
      Takashi Aoki, Toshinori Takahashi, Mika Tanda
    • Journal Title

      RIMS Kokyuroku Bessatsu

      Volume: B52 Pages: 165-174

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Six-dimensional Painleve systems and their particular solutions in terms of rigid systems2014

    • Author(s)
      Takao Suzuki
    • Journal Title

      J. Math. Phys.

      Volume: 55 Issue: 10 Pages: 102902-102902

    • DOI

      10.1063/1.4898766

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] Analytic continuation of multiple polylogarithm via residues2018

    • Author(s)
      Yayoi Nakamura
    • Organizer
      第11回ゼータ若手研究集会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] From Heine to q-Painleve2017

    • Author(s)
      Takao Suzuki
    • Organizer
      The 25th International Conference on Integrable Systems and Quantum symmetries
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Exact WKB analysis of the Gauss hypergeometric differential equation2017

    • Author(s)
      Takashi Aoki
    • Organizer
      RIMS-iTHEMS International Workshop on Resurgence Theory
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Exact WKB analysis of the Gauss hypergeometric differential equation2017

    • Author(s)
      Takashi Aoki
    • Organizer
      RIMS Joint Research Microlocal analysis and asymptotic analysis
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] A higher order generalization of the Painleve VI equation with W(A_{2n+1}^{(1)}) symmetry2017

    • Author(s)
      Takao Suzuki
    • Organizer
      Conformal field theory, isomonodromy tau-functions and Painleve equations
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Gauss hypergeometric function, the Kummer confluent hypergeo- metric function and WKB solutions2016

    • Author(s)
      Takashi Aoki
    • Organizer
      Resurgence at Kavli IPMU
    • Place of Presentation
      Kavli IPMU, University of Tokyo, Chiba
    • Year and Date
      2016-12-15
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The hypergeometric function, confluent hypergeometric functions and WKB solutions2016

    • Author(s)
      Takashi Aoki
    • Organizer
      RIMS Symposium on New development of microlocal analysis and singular perturbation theory
    • Place of Presentation
      RIMS, Kyoto University, Kyoto
    • Year and Date
      2016-10-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 合流型超幾何関数と WKB 解の関係2016

    • Author(s)
      青木貴史,反田美香,高橋甫宗
    • Organizer
      日本数学会
    • Place of Presentation
      関西大学, 大阪
    • Year and Date
      2016-09-15
    • Related Report
      2016 Research-status Report
  • [Presentation] A generalization of the q-Painleve VI equation from a viewpoint of a basic hypergeometric solution2016

    • Author(s)
      Takao Suzuki
    • Organizer
      Symmetries and Integrability of Difference Equations (SIDE) 12
    • Place of Presentation
      Sainte-Adele (Quebec, Canada)
    • Year and Date
      2016-07-08
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The hypergeometric function and WKB solutions2015

    • Author(s)
      Takashi Aoki
    • Organizer
      微分方程式の総合的研究
    • Place of Presentation
      東京大学数理科学研究科 東京
    • Year and Date
      2015-12-20
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] The hypergeometric function and WKB solutions2015

    • Author(s)
      Takashi Aoki
    • Organizer
      RIMS Symposium on Microlocal Analysis and Singular Perturbation Theory
    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto
    • Year and Date
      2015-10-09
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] The hypergeometric function and WKB solutions2015

    • Author(s)
      Takashi Aoki
    • Organizer
      Analytic, Algebraic and Geometric Aspects of Differential Equations
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2015-09-17
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Higher order Painleve systems, rigid systems and hypergeometric functions2015

    • Author(s)
      Takao Suzuki
    • Organizer
      Analytic, Algebraic and Geometric Aspects of Differential Equations
    • Place of Presentation
      Bedlewo, Poland
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 合流型超幾何函数とパンルヴェ方程式2015

    • Author(s)
      鈴木貴雄
    • Organizer
      函数方程式論サマーセミナー
    • Place of Presentation
      四季の湯強羅静雲荘 神奈川県足柄下郡箱根町
    • Year and Date
      2015-08-07
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] ドリンフェルト・ソコロフ階層とシンプソンのeven4超幾何関数2015

    • Author(s)
      鈴木貴雄
    • Organizer
      日本数学会2015年会
    • Place of Presentation
      明治大学(東京都千代田区)
    • Year and Date
      2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] 超幾何関数とWKB解2015

    • Author(s)
      反田美香, 青木貴史
    • Organizer
      日本数学会2015年会
    • Place of Presentation
      明治大学(東京都千代田区)
    • Year and Date
      2015-03-21
    • Related Report
      2014 Research-status Report
  • [Presentation] Exact WKB analysis of Schrdoeingier equation with a Stokes curve of loop type2015

    • Author(s)
      高橋甫宗, 岩木耕平, 青木貴史
    • Organizer
      日本数学会2015年会
    • Place of Presentation
      明治大学(東京都千代田区)
    • Year and Date
      2015-03-21
    • Related Report
      2014 Research-status Report
  • [Presentation] The hypergeometric function and WKB solutions2015

    • Author(s)
      Takashi Aoki
    • Organizer
      大阪梅田微分方程式 セミナー
    • Place of Presentation
      関西学院大阪梅田キャンパス(大阪府大阪市)
    • Year and Date
      2015-02-22
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Higher order Painleve systems, rigid systems and hypergeometric functions2014

    • Author(s)
      Takao Suzuki
    • Organizer
      RIMS Symposium on Recent developments in differential equations in the complex domain
    • Place of Presentation
      RIMS, Kyoto University(京都府京都市)
    • Year and Date
      2014-11-17
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] The hypergeometric function and WKB solutions2014

    • Author(s)
      Takashi Aoki
    • Organizer
      RIMS Symposium on Several aspects of micro local analysis
    • Place of Presentation
      RIMS, Kyoto University(京都府京都市)
    • Year and Date
      2014-10-20
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] アペル超幾何函数の高階化と2変数高階パンルヴェ方程式2014

    • Author(s)
      鈴木貴雄
    • Organizer
      函数方程式論サマーセミナー
    • Place of Presentation
      KKR伊豆長岡保養所「千歳荘」(静岡県伊豆の国市)
    • Year and Date
      2014-08-07
    • Related Report
      2014 Research-status Report
    • Invited
  • [Remarks] 近畿大学理工学部・総合理工学研究科 教員・研究紹介

    • URL

      http://www.kindai.ac.jp/sci/education/faculty_and_research/01_aoki_takashi.html

    • Related Report
      2016 Research-status Report
  • [Remarks] researchmap

    • URL

      http://researchmap.jp/read0034075

    • Related Report
      2016 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2022-02-16  

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