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Tasoev's continued fraction and its applications

Research Project

Project/Area Number 15540021
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KOMATSU Takao  Hirosaki University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (70300556)

Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥600,000 (Direct Cost: ¥600,000)
KeywordsContinued fractions / Tasoev continued fractions / Hurwitz continued fractions / Diophantine approximations / Rational approximations / Diophantine problems / Rogers-Ramanujan / Zaremba's conjecture / Rogers-Ramanujan continued fractions / Tasoev's continued fractions
Research Abstract

1.Definition and discovery of Tasoev continued fractions
Some basic types of examples of the Tasoev continued fractions were discovered and were well-defined. Applying the fact that Hurwitz continued fractions had three types of examples, namely, tanh-type, tan-type and e-type, the similar types of Tasoev continued fractions were constructed in a more general way, then in addition, the Hurwitz continued fractions corresponding to these general types were also discovered.
2.Relation between Tasoev continued fraction and other continued fractions
By finding that the Tasoev continued fraction and the Hurwitz continued fraction have a good comparison under the viewpoint that the sequence of partial quotients is geometric or arithmetic, respectively, once new results were obtained in either continued fraction, the corresponding new results were also obtained in another continued fraction. This structure will be applied in the future too. The Rogeres-Ramanujan continued fraction, one of the gen … More eral continued fractions whose numerators are not always 1,can be converted to the simple continued fraction in some cases, which becomes a Tasoev continued fraction. Some concrete examples were also given.
3.Rational approximation of Tasoev continued fraction
The way of evaluation so that the irrational number yielding Tasoev continued fraction is well-approximated by the rational number, was established. The exact values of such evaluations for some concrete Tasoev continued fractions were able to be given actually.
4.New applications of continued fraction
A new inhomogeneous continued fraction expansion algorithm was developed in the problem to obtain a value in inhomogeneous Diophantine problems. Leaping convergents, which are composed from every n-th convergents of the continued fraction, were defined, and their various interesting characters were found. The Zaremba's conjecture, which is a still open problem in finite continued fractions, were partially solved if the denominators of rational numbers had even powers. Less

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (52 results)

All 2005 2004 2003 Other

All Journal Article (46 results) Publications (6 results)

  • [Journal Article] On a Zaremba's conjecture for powers2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Sarajevo Journal of Mathematics 1(13)・1

      Pages: 9-13

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Rational approximations to Tasoev continued fractions, II2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Lietuvos Matematikos Rinkinys 45・1

      Pages: 84-94

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hurwitz and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Monatshefte fuer Mathematik 145・1

      Pages: 47-60

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An algorithm of infinite sums representations and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematics of Computation 74・252

      Pages: 2081-2094

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some recognizable forms of simple continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      International Journal of Mathematical Sciences 4・1

      Pages: 47-63

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a Zaremba's conjecture for powers2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Sarajevo Journal of Mathematics 1(13)

      Pages: 9-13

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Rational approximations to Tasoev continued fractions, II2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Lietuvos Matematikos Rinkinys 45-1

      Pages: 84-94

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Hurwitz and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Monatshefte fuer Mathematik 145-1

      Pages: 47-60

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An algorithm of infinite sums representations and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematics of Computation 74-252

      Pages: 2081-2094

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some recognizable forms of simple continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      International Journal of Mathematical Sciences 4-1

      Pages: 47-63

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On a Zaremba's conjecture for powers2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Sarajevo J.Math. 1(13)

      Pages: 9-13

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Rational approximations to Tasoev continued fractions, II2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Liet.Mat.Rink. 45

      Pages: 84-94

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Hurwitz and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Monatsh.Math. 145

      Pages: 47-60

    • Related Report
      2005 Annual Research Report
  • [Journal Article] An algorithm of infinite sums representations and Tasoev continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      Math.Comp. 74

      Pages: 2081-2094

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Some recognizable forms of simple continued fractions2005

    • Author(s)
      Takao Komatsu
    • Journal Title

      International J.Math.Sci. 4

      Pages: 47-63

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On inhomogeneous Diophantine approximation and the Borweins' algorithm2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Far East Journal of Mathematical Sciences (FJMS) 12・2

      Pages: 203-224

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arithmetic properties of the leaping convergents of e^(1/s)2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Tokyo Journal of Mathematics 27・1

      Pages: 1-12

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tasoev's continued fractions and Rogers-Ramanujan continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Journal of Number Theory 109・1

      Pages: 27-40

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Rational approximations to Tasoev continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematica Pannonica 15・2

      Pages: 199-207

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] 連分数から実数へ2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      数理解析研究所講究録 1384

      Pages: 15-23

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An algorithm of infinite sums' representations associated with Tasoev continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Bunyakovsky International Conference, Abstracts

      Pages: 164-164

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On inhomogeneous Diophantine approximation and the Borweins' algorithm2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Far East Journal of Mathematical Sciences (FJMS) 12-2

      Pages: 203-224

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arithmetic properties of the leaping convergents of e^(1/s)2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Tokyo Journal of Mathematics 27-1

      Pages: 1-12

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Tasoev's continued fractions and Rogers-Ramanujan continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Journal of Number Theory 109-1

      Pages: 27-40

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Rational approximationsto to Tasoev continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematica Pannonica 15-2

      Pages: 199-207

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] From continued fractions to real numbers (in Japanese)2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Suurikaisekikenkyushokokyuroku 1384

      Pages: 15-23

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] An algorithm of infinite sums' representations associated with Tasoev continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Bunyakovsky International Conference (Abstracts)

      Pages: 164-164

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On inhomogeneous Diophantine approximation and the Borweins' algorithm2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Far East J.Math.Sci.(FJMS) 7

      Pages: 203-224

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Arithmetic properties of the leaping convergents of e^<1/s>2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Tokyo J.Math. 27

      Pages: 1-12

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Tasoev's continued fractions and Rogers-Ramanujan continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      J.Number Theory 109

      Pages: 27-40

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Rational approximations to Tasoev continued fractions2004

    • Author(s)
      Takao Komatsu
    • Journal Title

      Math.Pannon. 15

      Pages: 199-207

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 連分数から実数へ2004

    • Author(s)
      小松 尚夫
    • Journal Title

      数理解析研究所講究録 1384

      Pages: 15-23

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The interval associated with a Fibonacci number2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Fibonacci Quarterly 41・1

      Pages: 3-6

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Tasoev's continued fractions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society 134・1

      Pages: 1-12

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Hurwitzian and Tasoev's continued fractions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Acta Arithmetica 107・2

      Pages: 161-177

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Simple continued fraction expansions of some values of certain hypergeometric functions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Tsukuba Journal of Mathematics 27・1

      Pages: 161-173

    • NAID

      120001139459

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the number of solutions of the Diophantine equation of Frobenius - General case2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematical Communications 8・2

      Pages: 195-206

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Recurrence relations of the leaping convergents2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      JP Journal of Algebra, Number Theory and Applications 3・3

      Pages: 447-459

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arithmetic properties of the leaping convergents of e^(1/s)2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      数理解析研究所講究録 1319

      Pages: 85-94

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The interval associated with a Fibonacci number2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Fibonacci Quarterly 41-1

      Pages: 3-6

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Tasoev's continued fractions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematical Proceedings of the Cambridge Philosophical Society 134-1

      Pages: 1-12

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Hurwitzian and Tasoev's continued fractions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Acta Arithmetica 107-2

      Pages: 161-177

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Simple continued fraction expansions of some values of certain hypergeometric functions2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Tsukuba Journal of Mathematics 27-1

      Pages: 161-173

    • NAID

      120001139459

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the number of solutions of the Diophantine equation of Frobenius-General case2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Mathematical Communications 8-2

      Pages: 195-206

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Recurrence relations of the leaping convergents2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      JP Journal of Number Theory, Algebra and Applications 3-3

      Pages: 447-459

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Arithmetic properties of the leaping convergents of e^(1/s)2003

    • Author(s)
      Takao Komatsu
    • Journal Title

      Suurikaisekikenkyushokokyuroku 1319

      Pages: 85-94

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] Takao Komatsu: "The interval associated with a Fibonacci number"Fibonacci Quarterly. 41・1. 3-6 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takao Komatsu: "On Tasoev's continued fractions"Mathematical Proceedings of the Cambridge Philosophical Society. 134・1. 447-459 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takao Komatsu: "On Hurwitzian and Tasoev's continued fractions"Acta Arithmetica. 107・2. 161-177 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takao Komatsu: "Simple continued fraction expansions of some values of certain hypergeometric functions"Tsukuba Journal of Mathematics. 27・1. 161-173 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takao Komatsu: "On the number of solutions of the Diophantine equation of Frobenius - General case"Mathematical Communications. 8・2. 195-206 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takao Komatsu: "Recurrence relations of the leaping convergents"JP Journal of Number Theory, Algebra and Applications. 3・3. 447-459 (2003)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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