Integrable system and middle convolution

• Project/Area Number: 22740107
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Chuo University
• Principal Investigator: TAKEMURA Kouichi 中央大学, 理工学部, 准教授 (10326069)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 可積分系 / 数理物理 / 複素領域の微分方程式 / ホインの微分方程式 / パンルヴェ方程式 / ミドルコンボルーション / 超離散化 / Inozemtsev模型 / オイラー積分変換 / みかけの特異点 / 積分表示 / 不確定特異点

Research Abstract

We investigated Heun's differential equation in the case that one of the sigularity is apparent. By applying middle convolution for it, we obtain an equation which admits a polynomial-type solution, and we have integral representations of solutions of the original differential equation. We also studied it with a connenction to generalized hypergeometric equations.
We obtained results on multi-indexed orthogonal polynomials and on ultradiscrete Painleve equations.

Research Products

• [Journal Article] Ultradiscrete Painlevé VI with Parity Variables (2013)
  • Author(s): Kouichi TAKEMURA, Terumitsu Tsutsui
  • Journal Title: SIGMA 9 Volume: 070 Pages: 1-12
  • DOI: 10.3842/sigma.2013.070
  • Peer Reviewed
• [Journal Article] Confluence of apparent singularities in multi-indexed orthogonal polynomials : the Jacobi case (2013)
  • Author(s): C.-L. Ho, Ryu Sasaki, Kouichi TAKEMURA
  • Journal Title: J. Phys Volume: A46 Issue: 11 Pages: 115205-115205
  • DOI: 10.1088/1751-8113/46/11/115205
  • Peer Reviewed
• [Journal Article] Euler's integral transformation for systems of linear differential equations with irregular singularities (2012)
  • Author(s): Kouichi TAKEMURA
  • Journal Title: Banach Center Publ Volume: 97 Pages: 141-152
  • DOI: 10.4064/bc97-0-10
  • Peer Reviewed
• [Journal Article] Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity SIGMA (2012)
  • Author(s): Ryu Sasaki, Kouichi TAKEMURA
  • Journal Title: SIGMA Volume: 8(085)
  • DOI: 10.3842/sigma.2012.085
  • Peer Reviewed
• [Journal Article] Source identity and kernel functions for Inozemtsev-type systems (2012)
  • Author(s): Edwin Langmann, Kouichi TAKEMURA
  • Journal Title: J. Math. Phys Volume: 53 Issue: 8 Pages: 82105-82105
  • DOI: 10.1063/1.4745001
  • Peer Reviewed
• [Journal Article] Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial (2012)
  • Author(s): Kouichi TAKEMURA
  • Journal Title: J. Phys Volume: A45 Issue: 8 Pages: 85211-85211
  • DOI: 10.1088/1751-8113/45/8/085211
  • Peer Reviewed
• [Journal Article] Introduction to middle convolution for differential equations with irregular singularities (2010)
  • Author(s): Kouichi TAKEMURA
  • Journal Title: New Trends in Quantum Integrable Systems Pages: 393-420
  • DOI: 10.1142/9789814324373_0020
  • Peer Reviewed
• [Journal Article] INTRODUCTION TO MIDDLE CONVOLUTION FOR DIFFERENTIAL EQUATIONS WITH IRREGULAR SINGULARITIES (2010)
  • Author(s): Kouichi TAKEMURA
  • Journal Title: NEW TRENDS IN QUANTUM INTEGRABLE SYSTEMS Pages: 393-420
  • Peer Reviewed
• [Presentation] Ultradiscrete Painlevé VI with Parity Variables (2013)
  • Author(s): 竹村剛一, 筒井栄光
  • Organizer: 日本数学会2013年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 愛媛大学
  • Year and Date: 2013-09-24
• [Presentation] Confuence of apparent singularities in multi-indexed orthogonal poly-nomials : the Jacobi case (2013)
  • Author(s): C.-L. Ho, 佐々木隆, 竹村剛一
  • Organizer: 日本数学会2013年度年会無限可積分系セッション
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-23
• [Presentation] Global solutions of certain second order differential equations with a high degree of apparent singularity (2013)
  • Author(s): 佐々木 隆, 竹村剛一
  • Organizer: 日本数学会2013年度年会函数方程式論分科会
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-20
• [Presentation] Ultradiscrete Painleve VI with parity variables (2013)
  • Author(s): 竹村 剛一，筒井 栄光
  • Organizer: 日本数学会２０１３年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 愛媛大学
• [Presentation] Global solutions of certain second order differential equations with a high degree of apparent singularity (2013)
  • Author(s): 佐々木 隆, 竹村 剛一
  • Organizer: 日本数学会２０１３年度年会函数方程式論分科会
  • Place of Presentation: 京都大学
• [Presentation] Confuence of apparent singularities in multi-indexed orthogonal poly-nomials: the Jacobi case (2013)
  • Author(s): Choon-Lin Ho, 佐々木 隆, 竹村 剛一
  • Organizer: 日本数学会２０１３年度年会無限可積分系セッション
  • Place of Presentation: 京都大学
• [Presentation] Source identity and kernel functions for Inozemtsev-type systems (2012)
  • Author(s): Edwin Langmann, 竹村剛一
  • Organizer: 日本数学会2012年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-18
• [Presentation] Source identity and kernel functions for Inozemtsev-type systems (2012)
  • Author(s): Edwin Langmann, 竹村 剛一
  • Organizer: 日本数学会２０１２年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 九州大学
• [Presentation] Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial (2011)
  • Author(s): 竹村剛一
  • Organizer: 日本数学会2011年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-30