New aspects of special functions

• Project/Area Number: 26400122
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Chuo University
• Principal Investigator: Takemura Kouichi 中央大学, 理工学部, 准教授 (10326069)
• Research Collaborator: Isojima Shin ; Igarashi Hikaru ; Kojima Kentaro ; Sato Tsukasa
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); 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: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ホインの微分方程式 / パンルヴェ方程式 / 超離散方程式 / ｑ差分 / q-Heun equation / Ruijsenaars system / 可積分系 / 特殊関数 / qホイン方程式 / アクセサリーパラメーター / Ruijsenaars系 / 退化 / q差分方程式 / Ruijsenaars 系 / 符号付き超離散化 / 離散パンルヴェ方程式 / 超離散 / エアリー関数 / 直交多項式 / ヤコビ多項式 / マヤ図形 / multi-indexed polynomial

Outline of Final Research Achievements

Special functions are the important function as well as the elememtary functions. We obtained several results on Heun's differential equation, Painleve equations and their q-deformations.
We investigated the asymptotics of the solutions to the ultradiscrete Painleve II equation. We introduced q-Heun equation and its variants by degerenations of the Ruijsenaars-van Diejen system. We also obtained the results on the relationship with the q-Painleve equations and on the characterization of the variants of q-Heun equation in terms of the difference analogue of the regular singularity and the apparent singularity.

Academic Significance and Societal Importance of the Research Achievements

超離散パンルヴェ第二方程式における成果は、ｑ離散パンルヴェ方程式やパンルヴェ方程式の解に対する研究への応用につながると考えている。また、q-ホイン方程式の研究は、Luc Vinet, Alexei Zhedanov らの別の動機からの研究にも関係しており、今後の発展につながる可能性がある。
潜在的には、新たな特殊関数として物理学などへの応用が期待される。

Research Products

• [Journal Article] On reducible monodromy representations of some generalized Lame equation (2018)
  • Author(s): Chen Zhijie、Kuo Ting-Jung、Lin Chang-Shou、Takemura Kouichi
  • Journal Title: Mathematische Zeitschrift Volume: 288 Issue: 3-4 Pages: 679-688
  • DOI: 10.1007/s00209-017-1906-z
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems (2018)
  • Author(s): Chen Zhijie、Kuo Ting-Jung、Lin Chang-Shou、Takemura Kouichi
  • Journal Title: Journal of Differential Equations Volume: 264 Issue: 8 Pages: 5408-5431
  • DOI: 10.1016/j.jde.2018.01.005
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On q-Deformations of the Heun Equation (2018)
  • Author(s): Takemura Kouichi
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: 14, 061 Pages: 1-16
  • DOI: 10.3842/sigma.2018.061
  • Peer Reviewed / Open Access
• [Journal Article] On two-parameter solutions of simultaneous ultradiscrete Painleve II equation with parity variables (2018)
  • Author(s): Igarashi Hikaru、Takemura Kouichi
  • Journal Title: Journal of Mathematical Physics Volume: 59 Issue: 10 Pages: 103502-103502
  • DOI: 10.1063/1.5020867
  • Peer Reviewed
• [Journal Article] Integral transformation of Heun's equation and some applications (2017)
  • Author(s): Kouichi Takemura
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 69 Issue: 2 Pages: 849-891
  • DOI: 10.2969/jmsj/06920849
  • NAID: 130006887125
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations (2017)
  • Author(s): Kouichi Takemura
  • Journal Title: J. Integrable Systems Volume: 2 Issue: 1
  • DOI: 10.1093/integr/xyx008
  • Peer Reviewed / Open Access
• [Journal Article] Integral transformation of Heun's equation and some applicationｓ (2017)
  • Author(s): Kouichi Takemura
  • Journal Title: J. Math. Soc. Japan Volume: 69
  • NAID: 130006887125
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] New Airy-type solutions of the ultradiscrete Painleve II equation with parity variables (2016)
  • Author(s): Hikaru Igarashi, Shin Isojima and Kouichi Takemura
  • Journal Title: J. Phys. A Volume: 49 Issue: 14 Pages: 145207-145207
  • DOI: 10.1088/1751-8113/49/14/145207
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Multi-indexed Jacobi polynomials and Maya diagrams (2014)
  • Author(s): Kouichi TAKEMURA
  • Journal Title: J. Math. Phys. Volume: 55 Issue: 11 Pages: 113501-113501
  • DOI: 10.1063/1.4899082
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] On q-deformations of the Heun equation (2018)
  • Author(s): 竹村 剛一
  • Organizer: 日本数学会2018年度秋季総合講演会無限可積分系セッション
• [Presentation] qホイン方程式の多項式解について (2018)
  • Author(s): 小嶋健太郎、佐藤司、竹村剛一
  • Organizer: 日本数学会2018年度年会無限可積分系セッション
• [Presentation] Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems (2018)
  • Author(s): Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin, 竹村剛一
  • Organizer: 日本数学会2018年度年会無限可積分系セッション
• [Presentation] Degenerations of Ruijsenaars-van Diejen operator and q-Painleve equations (2017)
  • Author(s): 竹村剛一
  • Organizer: 日本数学会2017年度秋季総合講演会無限可積分系セッション
• [Presentation] On solutions of ultradiscrete Painlave II equation with parity variables (2017)
  • Author(s): 五十嵐光, 竹村剛一
  • Organizer: 日本数学会2017年度秋季総合講演会函数方程式論分科会
• [Presentation] Degenerations of Ruijsenaars-van Diejen operator, q-Painlevé equations and q-Heun equations (2017)
  • Author(s): Kouichi Takemura
  • Organizer: Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics
  • Place of Presentation: Vienna, Austria
  • Int'l Joint Research / Invited
• [Presentation] New Airy-type solutions of the ultradiscrete Painleve II equation with parity variables (2016)
  • Author(s): Hikaru Igarashi, Shin Isojima and Kouichi Takemura
  • Organizer: 日本数学会2016年度年会函数方程式論分科会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
• [Presentation] Multi-indexed Jacobi polynomials and Maya diagrams (2014)
  • Author(s): 竹村 剛一
  • Organizer: 日本数学会2014年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25