Skip to main content
SpringerLink
Log in

On a two-valued sequence and related continued fractions in power series fields

The Ramanujan Journal volume 45, pages 859–871 (2018)Cite this article

Abstract

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over \(\mathbb {Q}\), having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set \(\lbrace 1,2\rbrace \). The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over \(\mathbb {F}_3\), which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Allouche, J.-P., Shallit, J.: Automatic Sequences. Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  2. Buck, M., Robbins, D.: The continued fraction expansion of an algebraic power series satisfying a quartic equation. J. Number Theory 50, 335–344 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Christol, G.: Ensembles presque périodiques \(k\) -reconnaissables. Theor. Comput. Sci. 9, 141–145 (1979)

    Article  MATH  Google Scholar 

  4. Christol, G., Kamae, T., Mendès, M., Mendès France, M., Rauzy, G.: Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108, 401–419 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lasjaunias, A.: Diophantine approximation and continued fraction expansions of algebraic power series in positive characteristic. J. Number Theory 65, 206–225 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lasjaunias, A.: A survey of diophantine approximation in fields of power series. Mon. für Math. 130, 211–229 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lasjaunias, A., Yao, J.-Y.: Hyperquadratic continued fractions and automatic sequences. Finite Fields Their Appl. 40, 46–60 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mahler, K.: On a theorem of Liouville in fields of positive characteristic. Can. J. Math. 1, 388–404 (1949)

    MathSciNet  MATH  Google Scholar 

  9. Mills, W., Robbins, D.P.: Continued fractions for certain algebraic power series. J. Number Theory 23, 388–404 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Roth, K.F.: Rational approximation to algebraic numbers. Mathematika 2, 1–20 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  11. Uchiyama, S.: Rational approximation to algebraic functions. Proc. Jpn. Acad. 36, 1–2 (1960)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux CNRS-UMR 5251, Université de Bordeaux, 33405, Talence, France

    Bill Allombert & Alain Lasjaunias

  2. CNRS, Laboratoire LIP (CNRS, ENSL, Inria, UCBL), Université de Lyon, ENS Lyon, 46 Allée d’Italie, 69364, Lyon Cedex 07, France

    Nicolas Brisebarre

Authors
  1. Bill Allombert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolas Brisebarre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alain Lasjaunias
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Lasjaunias.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allombert, B., Brisebarre, N. & Lasjaunias, A. On a two-valued sequence and related continued fractions in power series fields. Ramanujan J 45, 859–871 (2018). https://doi.org/10.1007/s11139-017-9892-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-017-9892-7

Keywords

Mathematics Subject Classification

search

Navigation