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THEORY OF MAHLER FUNCTIONS AND APPLICATIONS TO REGULAR SEQUENCES

Research Project

Project/Area Number 09640060
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKEIO UNIVERSITY

Principal Investigator

NISHIOKA Kumiko  KEIO UNIV., ECONOMICS,PROF., 経済学部, 教授 (80144632)

Co-Investigator(Kenkyū-buntansha) NISHIOKA Keiji  KEIO UNIV., ENV.INFO., PROF., 環境情報学部, 教授 (10228158)
KOMIYA Hidetoshi  KEIO UNIV., COMMERCE,PROF., 商学部, 教授 (90153676)
WATABE Mutsuo  KEIO UNIV., COMMERCE,PROF., 商学部, 教授 (30080493)
HIKARI Michitaka  KEIO UNIV., ECONOMICS,PROF., 経済学部, 教授 (30056296)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1998: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1997: ¥1,500,000 (Direct Cost: ¥1,500,000)
Keywordstranscendental number / algebraic independence / Mahler function / Fibonacci number / Theta series / finite automaton / フィボナッチ数 / テ-タ級数
Research Abstract

If a function f(z) satisfies a functional equation under the trasformation z <does not result in>z^d, f(z) is called a Mahler function. The transcendence and the algebraic independence of the values of Mauler functions have been studied for a long time. Until recently we have satisfactory results only in the case Mahler functions satisfy functional equations under a common trasformation z <does not result in>z^d. If we simultaneously treate Mahler functions under various transformations, a lot of difficult problems occur. Our research provides a solution to these problems. First we have obtained a widely applicable theorem on algebraic independence. Second we applied it to algebraic independence of reciprocal sums of Fibonacci numbers. This result gives an almost complete solution to a problem proposed by Erdos and Graham. We also proved that 2-adic expansions of real algebraic numbers can not be represented by finte automata, which means that 2-adic expansions of real algebraic numbers are complex in a certain sense.
On the other hand, applying Nesterenko's theorem on the algebraic independence of the values of modular functions, we obtained a transcendence result of the reciprocal sums of Fibonacci numbers which can not be proved by using Mahler functions. We also obtained the transcendence of Theta series and Rogers-Ramanujan continued fraction.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report

Research Products

(29 results)

All Other

All Publications (29 results)

  • [Publications] Kumiko Nishioka: "Algebraic independence of reciprocal sums of binary recurrence" Mh.Math.123. 135-148 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Daniel Duverney: "Transcendence of Rogers-Ramanujancontinued fraction and Reciprocal sums of Fibonacci numbers" Proc.Japan Academy. 73. 140-142 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Daniel Duverney: "Transcendence of Jacobils theta series and related results" Number Theory, Eds.by Gyory 他. 157-168 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka: "Algebraic independence of sums of reciprocals of the Fibonacci numbers" Math.Nachr.(予定). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka: "Substitusion in two symbols and transcendence" Tokyo J.Math.(予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka: "Mahler functions and binary linear recurrence sequences" 日仏超越数論研究集会報告集. (予定). (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Keiji Nishioka: "Lie extensions" Proc.Japan Academy. 73. 82-85 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Michitaka Hikari: "Simple Skew linear groups of Degree 3" 〓iyoshi Review of Natural Science, Keio Univ.21. 23-27 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Michitaka Hikari: "Skew linear groups of odd order" Hiyoshi Review of Natural Science, Keio Univ.23. 1-4 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka: ""Algebraic independence of reciprocal sums of binary recurrences"" Mh.Math.123. 135-148 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Daniel Duverney et al.: ""Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers"" Proc.Japan Academy. 73. 140-142 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Daniel Duverney et al.: ""Transcendence of Jacobi's theta series and related results"" Number Theory, eds. : Gyory et al.157-168 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka et al.: ""Algebraic independence of sums of reciprocals of Fibonacci numbers"" Math.Nachr.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka et al.: ""Substitution in two symbols and transcendence"" Tokyo J.Math.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Kumiko Nishioka: ""Mahler functions and binary linear recurrences"" Proc.France-Japan conference of transcendental number theory.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Keiji Nishioka :""Lie extensions"" Proc.Japan Academy. 73. 82-85 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Michitaka Hikari: ""Simple skew linear groups of degree 3"" Hiyoshi Review of Natural Science, Keio Univ.21. 23-27 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Michitaka Hikari: ""Skew linear groups of odd order"" Hiyoshi Review of Natural Science, Keio Univ.23. 1-4 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Daniel Duverney: "Transcendence of Jacobi's theta series and related results" Number Theory,Eds.by Gyory 他. 157-256 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kumiko Nishioka: "Algebraic independence of sums of reciprocals of the Fibonacci numbers" Math.Nachr.(予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Kumiko Nishioka: "Substitutions in two symbols and transcendence" Tokyo J.of Math.(予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Kumiko Nishioka: "Mahler functions and binary linear recurrence sequences" 日仏超越数論研究集会報告集. (予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] Michitaka Hikari: "Skew linear groups of odd order" Hiyoshi Review of Natural Science,Keio Univ.23. 1-4 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Daniel Duverney: "Transcendence of Rogers-Remanujan continued fraction and reciprocal sums of Fibonacci numbers" Proc.Japan Academy. 73・7. 140-142 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kumiko Nishioka: "Transcendence of Jacobi's theta series and related results" Proc.Conf.Number Theory Eger. (予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Kumiko Nishioka: "Algebraic independence of sums of reciprocals of the Fibonacci numbers"

    • Related Report
      1997 Annual Research Report
  • [Publications] Math.Nachr.(予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Kumiko Nishioka: "Substitutions in two symbols and transcendence" Tokyo J.of Math.(予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] Keiji Nishioka: "Lie extensions" Proc.Japan Academy. 73・5. 82-85 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1997-03-31   Modified: 2016-04-21  

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