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A Research on Linear Independence of Special Values of q-Functions

Research Project

Project/Area Number 13640007
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionGunma University

Principal Investigator

AMOU Masaaki  Gunma University, Faculty of Engineering. Associate Professor, 工学部, 助教授 (60201901)

Co-Investigator(Kenkyū-buntansha) AMANO Kazuo  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90137795)
IKEHATA Masaru  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 教授 (90202910)
SAITOU Saburho  Gunma University, Faculty of Engineering. Professor, 工学部, 教授 (10110397)
KATSURADA Masanori  Keio University, Faculty of Economics, Professor, 経済学部, 教授 (90224485)
SATO Hisashi  Gunma University, Faculty of Engineering, Assistant, 工学部, 助手 (60008513)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,000,000 (Direct Cost: ¥3,000,000)
Fiscal Year 2002: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Keywordsq- Function / q-Hypergeometric Sereis / Irrationality Measure / Linear Independence / Rationality Criterion / Theta Series / Linear Recurrence Sequence
Research Abstract

We have studied linear independence of special values of certain q-functions and obtained the following results.
1. We proved linear independence of the values of functions satisfying linear second order q-difference equations. The head investigator published a paper on the result (Proceedings of the China-Japan Seminar, in press) with Prof. T. Matala-aho (University of Oulu).
2. We considered the values of power series introduced by Bezivin and proved linear independence of them qualitatively, which generalized earlier results obtained by Bezivin. The head investigator published a paper on the result (The Ramanujan J., in press) with Prof. K. Vaananen (University of Oulu, a investigator from abroad).
3. We considered the values of generalized Tschakaloff series and obtained quantitative irrationality and linear independence results. The head investigator published a paper on the result (J. Number Theory, in press) with Prof. M. Katsurada (Keio University, one of the investigators).
4. We proved linear independence of the values of theta series. The head investigator submitted a paper on the result with Prof. Vaananen.
By the support of the grant concerned, the head investigator visited Prof. Vaananen and Prof. Matala-aho twice (August-September, 2001 ; August-September, 2002), which was very important for the present research.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (20 results)

All Other

All Publications (20 results)

  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory ; the joint Proceedings of the China-Japan Number Theory Conference. 17-25 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, K.Vaananen: "Arithmetic properties of solutions of certain functional equations in several variables"J.Number Theory. 96. 89-104 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, T.Matala-aho: "Arithmetic properties of solutions of linear second order q-difference equations"Number Theoretic Methods-Future Trends ; the Proceedings of the China-Japan Number Theory Conference. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, K.Vaananen: "Linear independence of the values of q-hypergeometric series and related functions"The Ramanujan J.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, K.Vaananen: "An analogue of Shidlovskii's lemma for certain q-difference equations"Aequations Math.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada: "Irrationality results for values of generalized Tschakaloff series II"J.Number Theory. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou, M. Katsurada, and K. Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory ; the joint Proceedings of the China-Japan Number Theory Conference. 17-25 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou and K. Vaananen: "Arithmetic properties of solutions of certain functional equations in several variables"J. Number Theory. 96. 89-104 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou and K. Vaananen: "Linear independence of the values of q-hypergeometric series and related functions"Number Theoretic Methods - Future Trends ; the Proceedings of the China-Japan Number Theory Conference. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou and K. Vaananen: "An analogue of Shidlovskii's lemma for certain q-difference equations"The Ramanujan J.. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou and T. Matala-aho: "Arithmetical properties of solutions of linear second order q-difference equations"J. Number Theory. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Amou and M. Katsurada: "Irrationality results for values of generalized Tschakaloff series II"Aequationes Math. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Amou, M.Katsurada, K.Vaananen: "On the values of certain q-hypergeometric series II"Analytic Number Theory ; the joint Proceedings of the China-Japan Number Theory Conference. 17-25 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Amou, K.Vaananen: "Arithmetic properties of solutions of certain functional equations in several variables"J. Number Theory. 96. 89-104 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Amou, T.Matala-Aho: "Arithmetical properties of solutions of linear second order q-difference equations"Number Theoretic Methods-Future Trends ; the Proceedings of the China-Japan Number Theory Conference. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Amou, K.Vaananen: "Linear independence of the values of q-hypergeometric series and related functions"The Ramanujan J.. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Amou, M.Katsurada: "Irrationality results for values of generalized Tschakaloff series II"J. Number Theory. (印刷中).

    • Related Report
      2002 Annual Research Report
  • [Publications] M. Amou, M. Katsurada, K. Va^^¨a^^¨na^^¨nen: "Arithmetical properties of the values of functions satisfying certain functional equations of Poincare^´"Acta Arith.. 99・4. 389-407 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Amou, K. Va^^¨a^^¨na^^¨nen: "Arithmetical properties of solutions of certain functional equations in several variables"J. Number Theory. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] M. Amou, K. Va^^¨a^^¨na^^¨nen: "An analogue of Shidlovskii's lemma for certain q-difference equations"Aequationes Math.. (印刷中).

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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