Discretization of higher order Painleve system and rigid system

• Project/Area Number: 15K04911
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kindai University
• Principal Investigator: SUZUKI Takao 近畿大学, 理工学部, 准教授 (60527208)
• Research Collaborator: Okubo Naoto 青山学院大学
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: パンルヴェ方程式 / 離散可積分系 / リー代数 / クラスター代数 / ワイル群 / 超幾何函数 / 超幾何関数

Outline of Final Research Achievements

Recently higher order generalizations of the Painleve equations are proposed both on continuous side and on discrete side. However we haven't clarified yet a classification of equations or a relationship with hypergeometric functions. In this work we have formulated birational representations of a reducible extended affine Weyl group with the aid of cluster mutations. Translations of this group provide the known higher order q-Painleve equations containing the q-hypergeometric functions as particular solutions.

Academic Significance and Societal Importance of the Research Achievements

高階パンルヴェ方程式の分類問題は，2階の場合と比べて初期値空間の代数幾何が難しくなることもあって，未だ有効な手段が発見されていない．微分の場合にはKatz，原岡，大島らによって確立された分類理論が存在するが，差分の場合については未だ想像もつかない．本研究の結果は，この問題を解決するための一つのきっかけとなるかもしれない．また，クラスター代数との関連が明らかにされたことで，正準量子化や共形場理論の方面への更なる発展も期待出来る．

Research Products

• [Journal Article] A Higher Order Painlev&eacute; System in Two Variables and Extensions of the Appell Hypergeometric Functions <i>F</i>₁, <i>F</i>₂ and <i>F</i>₃ (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
  • Peer Reviewed / Open Access
• [Journal Article] A reformulation of the generalized $q$-Painlev\'{e} VI system with $W(A^{(1)}_{2n+1})$ symmetry (2017)
  • Author(s): Takao Suzuki
  • Journal Title: J. Integrable Syst. Volume: 2 Issue: 1 Pages: 1-18
  • DOI: 10.1093/integr/xyw017
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] A q-analogue of the Drinfeld-Sokolov hierarchy of type A and q-Painleve system (2015)
  • Author(s): Takao Suzuki
  • Journal Title: AMS Contemp. Math. Volume: 651 Pages: 25-38
  • Peer Reviewed
• [Presentation] $A_{2n+1}^{(1)}$ 型 $q$-ドリンフェルト・ソコロフ階層の相似簡約と $q$-ガルニエ系 (2019)
  • Author(s): 鈴木 貴雄, 大久保 直人
  • Organizer: 日本数学会年会・無限可積分系セッション
• [Presentation] クラスター代数と高階 $q$-パンルヴェ系 (2018)
  • Author(s): 鈴木 貴雄
  • Organizer: RIMS共同研究（公開型）「可積分系理論から見える数理構造とその応用」
  • Invited
• [Presentation] 自然数の分割 $(n+1,n+1,n+1)$ に対応する $A$ 型ドリンフェルト・ソコロフ階層の相似簡約 (2018)
  • Author(s): 鈴木 貴雄
  • Organizer: 日本数学会秋季総合分科会・無限可積分系セッション
• [Presentation] Cluster algebra and generalized $q$-Painlev\'e VI systems of type $A$ (2018)
  • Author(s): Takao Suzuki
  • Organizer: Symmetries and Integrability of Difference Equations (SIDE) 13
  • Int'l Joint Research / Invited
• [Presentation] Cluster algebra and generalized $q$-Painlev\'e VI systems of type $A$ (2018)
  • Author(s): Takao Suzuki
  • Organizer: Conformal field theory, isomonodromy tau-functions and Painlev\'e equations, 2018
  • Int'l Joint Research / Invited
• [Presentation] From Heine to $q$-Painlev\'e (2017)
  • Author(s): Takao Suzuki
  • Organizer: The 25th International Conference on Integrable Systems and Quantum symmetries
  • Int'l Joint Research
• [Presentation] $A$型パンルヴェ系と超幾何函数 (2017)
  • Author(s): 鈴木貴雄
  • Organizer: RIMS共同研究（公開型）「可積分系の数理と応用」
  • Invited
• [Presentation] $W(A^{(1)}_{2n+1})$-対称性を持つ一般化 $q$-パンルヴェVI方程式に付随する双線型関係式 (2017)
  • Author(s): 鈴木貴雄
  • Organizer: 日本数学会秋季総合分科会・無限可積分系セッション
• [Presentation] A higher order generalization of the Painlev\'e VI equation with $W(A_{2n+1}^{(1)})$ symmetry (2017)
  • Author(s): Takao Suzuki
  • Organizer: Conformal field theory, isomonodromy tau-functions and Painlev\'e equations
  • Int'l Joint Research
• [Presentation] From Heine to $q$-Painlev\'e (2016)
  • Author(s): Takao Suzuki
  • Organizer: Integrable Systems 2016
  • Place of Presentation: The University of Sydney (Australia)
  • Year and Date: 2016-12-02
  • Int'l Joint Research
• [Presentation] $q$-パンルヴェVI方程式の $q$-超幾何関数の観点からの一般化 (2016)
  • Author(s): 鈴木貴雄
  • Organizer: 非線形波動研究の深化と展開
  • Place of Presentation: 九州大学応用力学研究所 (福岡県春日市)
  • Year and Date: 2016-11-04
• [Presentation] $q$-パンルヴェVI方程式の $q$-超幾何関数の観点からの一般化 (2016)
  • Author(s): 鈴木貴雄
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: いこいの村能登半島 (石川県羽咋郡志賀町)
  • Year and Date: 2016-08-05
• [Presentation] A generalization of the $q$-Painlev\'e VI equation from a viewpoint of a basic hypergeometric solution (2016)
  • 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
  • Int'l Joint Research
• [Presentation] q-超幾何関数 3φ2 を解に持つ4階 q-パンルヴェ方程式 (2016)
  • Author(s): 鈴木貴雄
  • Organizer: 日本数学会年会・無限可積分系セッション
  • Place of Presentation: 筑波大学(茨城県つくば市)
  • Year and Date: 2016-03-18
• [Presentation] Higher order Painleve systems, rigid systems and hypergeometric functions (2015)
  • Author(s): Takao Suzuki
  • Organizer: Analytic, Algebraic and Geometric Aspects of Differential Equations
  • Place of Presentation: The Mathematical Research and Conference Center, Bedlewo (Poland)
  • Year and Date: 2015-09-14
  • 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