Theory and numerical evaluation of special functions of several variables

• Project/Area Number: 25287018
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kobe University
• Principal Investigator: Takayama Nobuki 神戸大学, 理学(系)研究科(研究院), 教授 (30188099)
• Co-Investigator(Kenkyū-buntansha): 小池 達也 神戸大学, 理学(系)研究科(研究院), 准教授 (80324599)
• Co-Investigator(Renkei-kenkyūsha): HARAOKA Yoshishige 熊本大学, 大学院先端科学研究部(理), 教授 (30208665) ; MATSUMOTO Keiji 北海道大学, 理学研究院, 教授 (30229546)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥6,630,000 (Direct Cost: ¥5,100,000、Indirect Cost: ¥1,530,000); Fiscal Year 2015: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000); Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000); Fiscal Year 2013: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
• Keywords: 超幾何関数 / 数値評価 / 多変数超幾何関数 / Borel 総和 / 多変数超幾何多項式 / A超幾何系 / 合流型 A-超幾何関数 / ホロノミック勾配法 / order polytope / Borel変換

Outline of Final Research Achievements

We give an algorithm to evaluate numerically big A-hypergeometric polynomials. An efficient implementation to evaluate numerically the matrix hypergeometric function 1F1 is given. We study divergent series solutions of A-hypergeometric equations and those of Heun type ordinary differential equations in terms of the Borel summability. These theoretical results help to analyze these functions globally and we hope that they give a new method for a numerical analysis of these functions.

Research Products

• [Int'l Joint Research] University of Sevilla(Spain)
• [Journal Article] Pfaffian Systems of A-Hypergeometric Equations I, Bases of Twisted Cohomology Groups (2017)
  • Author(s): T.Hibi, K.Nishiyama, N.Takayama
  • Journal Title: Advances in Mathematics Volume: 306 Pages: 303-327
  • DOI: 10.1016/j.aim.2016.10.021
  • Peer Reviewed
• [Journal Article] Irregularity of Modified $A$-Hypergeometric Systems (2015)
  • Author(s): F.Castro-Jimenez, M.C.Fernandez-Fernandez, T.Koike, N.Takayama
  • Journal Title: Transaction of AMS Volume: 367 Issue: 8 Pages: 5415-5445
  • DOI: 10.1090/s0002-9947-2014-06225-9
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The singular locus of Lauricella&apos;s <i>F</i>_<i>C</i> (2014)
  • Author(s): Ryohei Hattori, Nobuki Takayama
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 66 Issue: 3 Pages: 981-995
  • DOI: 10.2969/jmsj/06630981
  • NAID: 130004927753
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Holonomic Rank of the Fisher-Bingham System of Differential Equations (2014)
  • Author(s): T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama
  • Journal Title: Journal of Pure and Applied Algebra Volume: 218 Issue: 11 Pages: 2060-2071
  • DOI: 10.1016/j.jpaa.2014.03.004
  • Peer Reviewed
• [Journal Article] On the Borel summability of 0-parameter solutions of nonlinear ordinary differential equations (2013)
  • Author(s): S. Kamimoto and T. Koike
  • Journal Title: Kokyuroku Bessatsu Volume: B40 Pages: 191-212
  • Peer Reviewed
• [Journal Article] Exact WKB analysis of second-order non-homogeneous linear ordinary differential equations (2013)
  • Author(s): T. Koike and Y.Takei
  • Journal Title: Kokyuroku Bessatsu Volume: B40 Pages: 293-312
  • Peer Reviewed
• [Presentation] A-超幾何関数のみたすPfaffian方程式系の構成 (2017)
  • Author(s): 高山信毅
  • Organizer: 北海道特殊関数セミナー
  • Place of Presentation: 北海道大学
  • Invited
• [Presentation] 完全 WKB 解析と multisummability について (2016)
  • Author(s): 小池達也
  • Organizer: 超幾何方程式研究会 2016
  • Place of Presentation: 神戸大学(兵庫県、神戸市)
  • Year and Date: 2016-01-07
• [Presentation] A-超幾何多項式の近似値公式と分割表統計 (2016)
  • Author(s): 高山信毅
  • Organizer: 超幾何方程式研究会 2016
  • Place of Presentation: 神戸大学(兵庫県、神戸市)
  • Year and Date: 2016-01-06
• [Presentation] A remark on the growth order of Borel transform of WKB solutions of one-dimensional Schr\"{o}dinger equations --- Toward a proof of its multisummability (2015)
  • Author(s): Tatsuya Koike
  • Organizer: 超局所解析と特異摂動論
  • Place of Presentation: 京都大学数理解析研究所(京都府、京都市)
  • Year and Date: 2015-10-09
  • Int'l Joint Research / Invited
• [Presentation] Groebner bases and holonomic gradient method (2015)
  • Author(s): Nobuki Takayama
  • Organizer: The 8th Mathematical Society of Japan Seasonal Institute, Current Trends on Gr\"obner Bases
  • Place of Presentation: Hotel Nikko Osaka (大阪府、大阪市)
  • Year and Date: 2015-07-07
  • Int'l Joint Research
• [Presentation] Sylvester 型行列による超幾何関数の数値評価 --- グレブナー基底を使わない方法 (2015)
  • Author(s): 高山信毅
  • Organizer: Risa/Asir Conference 2015
  • Place of Presentation: 金沢大学
  • Year and Date: 2015-03-18
• [Presentation] 超幾何多項式の期待値写像と情報幾何 (2015)
  • Author(s): 高山信毅
  • Organizer: 超幾何方程式研究会 2015
  • Place of Presentation: 神戸大学
  • Year and Date: 2015-01-07
• [Presentation] ホロノミック系の数値解析の統計への応用 (2014)
  • Author(s): 高山信毅
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学
  • Year and Date: 2014-12-21
  • Invited
• [Presentation] 多変数超幾何関数の数値計算とその応用, (2014)
  • Author(s): 高山信毅
  • Organizer: 第八回玉原特殊多様体研究集会
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2014-09-17
• [Presentation] Introduction and recent progress of the holonomic gradient method (2014)
  • Author(s): Nobuki Takayama
  • Organizer: CASTA 2014
  • Place of Presentation: 京都
  • Invited
• [Book] 超幾何学校2013/2014講義録, Rokko Lectures in Mathematics 23. (2015)
  • Author(s): 高山信毅(編)
  • Total Pages: 134
  • Publisher: 神戸大学大学院理学研究科数学専攻