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Periodica Mathematica Hungarica

, Volume 55, Issue 1, pp 35–59 | Cite as

Independence of continued fractions in the field of Laurent series

  • Tuangrat ChaichanaEmail author
  • Vichian Laohakosol
Article
  • 58 Downloads

Abstract

Criteria for independence, both algebraic and linear, are derived for continued fraction expansions of elements in the field of Laurent series. These criteria are then applied to examples involving elements recently discovered to have explicit series and continued fraction expansions.

Key words and phrases

independence continued fractions field of Laurent series 

Mathematics subject classification number

11A55 11F70 

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References

  1. [1]
    W. W. Adams, The algebraic independence of certain Liouville continued fractions, Proc. Amer. Math. Soc., 95 (1985), 512–516.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Baker, Continued fractions of transcendental numbers, Mathematika, 9 (1962), 1–8.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann., 96 (1932), 367–377.CrossRefGoogle Scholar
  4. [4]
    P. Bundschuh, Fractions continues et indépendance algébrique en p-adique, Astérisque, 41–42 (1977), 179–181.MathSciNetGoogle Scholar
  5. [5]
    P. Bundschuh, p-adische Kettenbrüche und Irrationalität Irrationalität p-adischer Zahlen, Elem. Math., 32 (1977), 36–40.zbMATHMathSciNetGoogle Scholar
  6. [6]
    P. Bundschuh, Über eine Klasse reeller transzendenter Zahlen mit explizit angebbarer g-adischer und Kettenbruch-Entwicklung, J. Reine Angew. Math., 318 (1980), 110–119.zbMATHMathSciNetGoogle Scholar
  7. [7]
    P. Bundschuh, Transcendental continued fractions, J. Number Theory, 18 (1984), 91–98.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    T. Chaichana, V. Laohakosol and A. Harnchoowong, Linear independence of continued fractions in the field of formal series over a finite field, Thai J. Math., 4 (2006), 163–177.MathSciNetzbMATHGoogle Scholar
  9. [9]
    L. V. Danilov, Some classes of transcendental numbers, Mat. Zametki, 12 (1972), 149–154, in Russian: English transl., Math. Notes, 12(1972), 524–527.zbMATHMathSciNetGoogle Scholar
  10. [10]
    J. Hančl, Linear independence of continued fractions, J. Théor. Nombres Bordeaux, 14 (2002), 489–495.zbMATHMathSciNetGoogle Scholar
  11. [11]
    A. Kacha, Fractions continues p-adiques et indépendance algébrique, Bull. Belg. Math. Soc. Simon Stevin, 6 (1999), 305–314.zbMATHMathSciNetGoogle Scholar
  12. [12]
    V. Laohakosol and P. Ubolsri, Some algebraically independent continued fractions, Proc. Amer. Math. Soc., 95 (1985), 169–173.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    V. Laohakosol and P. Ubolsri, p-adic continued fractions of Liouville type, Proc. Amer. Math. Soc., 101 (1987), 403–410.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. Lianxiang, p-adic continued fractions I, Scientia Sinica (Ser. A), 28 (1985), 1009–1017.Google Scholar
  15. [15]
    W. Lianxiang, p-adic continued fractions II, Scientia Sinica (Ser. A), 28 (1985), 1018–1022.Google Scholar
  16. [16]
    W. Lianxiang, p-adic continued fractions III, Acta Math. Sinica (N.S.), 2 (1986), 299–308.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    K. Mahler, On a theorem of Liouville in fields of positive characteristic, Canad. J. Math., 1 (1949), 397–400.zbMATHMathSciNetGoogle Scholar
  18. [18]
    G. Nettler, Transcendental continued fractions, J. Number Theory, 13 (1981), 456–462.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    P. Riyapan, V. Laohakosol and T. Chaichana, Two types of explicit continued fractions, Period. Math. Hungar., 52 (2006), 1–22.CrossRefMathSciNetGoogle Scholar
  20. [20]
    W. M. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith., 95 (2000), 139–166.zbMATHMathSciNetGoogle Scholar
  21. [21]
    T. Töpfer, On the transcendence and algebraic independence of certain continued fractions, Monatsh. Math., 117 (1994), 255–262.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    G. C. Webber, Transcendence of certain continued fractions, Bull. Amer. Math. Soc., 50 (1944), 736–740.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsChulalongkorn UniversityBangkokThailand
  2. 2.Department of MathematicsKasetsart UniversityBangkokThailand

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