Remarkable algebraic independence property of certain series related to continued fractions

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove, using Mahler's method, the following results: Theorem 1 asserts that the series Θx,a,q) are algebraically independent for any distinct triplets (x,a,q) of nonzero algebraic numbers, where Θ (x,a,q) has the property shown in Corollary 1 that Θ (a,a,q) is expressed as a continued fraction. Theorem 2 asserts, under the weaker condition than that of Theorem 1, that the values Θ(x,1,q) are algebraically independent for any distinct pairs (x,q) of nonzero algebraic numbers. Typical examples of these results are generated by Fibonacci numbers.

Original languageEnglish
Title of host publicationDiophantine Analysis and Related Fields, DARF 2007/2008
Pages190-204
Number of pages15
DOIs
Publication statusPublished - 2008 Dec 1
EventDiophantine Analysis and Related Fields, DARF 2007/2008 - Kyoto, Japan
Duration: 2008 Mar 52008 Mar 7

Publication series

NameAIP Conference Proceedings
Volume976
ISSN (Print)0094-243X
ISSN (Electronic)1551-7616

Other

OtherDiophantine Analysis and Related Fields, DARF 2007/2008
CountryJapan
CityKyoto
Period08/3/508/3/7

Keywords

  • Algebraic independence
  • Continued fractions
  • Fibonacci numbers
  • Mahler's method

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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  • Cite this

    Tanaka, T. A. (2008). Remarkable algebraic independence property of certain series related to continued fractions. In Diophantine Analysis and Related Fields, DARF 2007/2008 (pp. 190-204). (AIP Conference Proceedings; Vol. 976). https://doi.org/10.1063/1.2841905