Structural analysis of transcendental numbers through the values of Mahler functions

2012 Fiscal Year Final Research Report

• Project/Area Number: 22740023
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: Keio University
• Principal Investigator: TANAKA Takaaki 慶應義塾大学, 理工学部, 専任講師 (60306850)
• Project Period (FY): 2010 – 2012
• Keywords: Mahler 関数 / 代数的独立性 / 整関数 / 無限積 / フィボナッチ数列 / 超越数

Research Abstract

In this project the research representative constructed, using linear recurrences, entire functions defined by explicit infinite products such that their values as well as their all successive derivatives at algebraic points other than their zeroes are al

Research Products

• [Journal Article] Algebraic relations with the infinite products generated by Fibonacci numbers (2013)
  • Author(s): T. Kurosawa, Y. Tachiya, and T. Tanaka
  • Journal Title: Ann. Math. Inform Volume: 41(掲載決定)
  • Peer Reviewed
• [Journal Article] Algebraic independence properties related to certain infinite products, AIP Conference Proceedings 1385,Diophantine Analysis and Related Fields 2011 (2012)
  • Author(s): T. Tanaka
  • Journal Title: American Institute of Physics Pages: 116-123
  • URL: http://dx.doi.org/10.1063/1.3630047
• [Journal Article] Algebraic independence of infinite products generated by Fibonacci numbers (2010)
  • Author(s): T. Kurosawa, Y. Tachiya, and T. Tanaka
  • Journal Title: Tsukuba J. Math Volume: 34 Pages: 255-264
  • URL: http://projecteuclid.org/euclid.tkbjm/1302268248
  • Peer Reviewed
• [Presentation] Algebraic independence of the values of certain infinite products and their derivatives related to Fibonacci and Lucas numbers (2012)
  • Author(s): T. Tanaka
  • Organizer: RIMS 研究集会「解析的整数論とその周辺 近似と漸近的手法を通して見た数論」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-10-30
• [Presentation] Algebraic independence of the values of Mahler functions with 'negative'transformations (2012)
  • Author(s): T. Tanaka
  • Organizer: Diophantine Analysis and Related Fields 2012
  • Place of Presentation: 新潟大学
  • Year and Date: 2012-01-10
• [Presentation] Algebraic independence properties related to certain infinite products and Lambert series (2011)
  • Author(s): T. Tanaka
  • Organizer: Diophantine Analysis and Related Fields 2011
  • Place of Presentation: 成蹊大学
  • Year and Date: 2011-03-03
• [Presentation] Algebraic independence of the values of a certain entire function defined by the infinite product (2010)
  • Author(s): T. Tanaka
  • Organizer: RIMS 研究集会「解析数論 複素関数の値の分布と性質を通して」
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-10-08