Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

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Abstract

The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a, q) at any distinct algebraic points to be algebraically independent, where Θ(x,a, q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a, q) taking algebraically independent values for any distinct triplets (x,a, q) of nonzero algebraic numbers. Moreover, Θ(a,a, q) is expressed as an irregular continued fraction and Θ(x, 1, q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.

Original languageEnglish
Pages (from-to)3081-3093
Number of pages13
JournalJournal of Number Theory
Volume129
Issue number12
DOIs
Publication statusPublished - 2009 Dec

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Algebraic Independence
Linear Recurrence
Continued fraction
Corollary
Analogue
Distinct
Hypergeometric Series
Lame number
Series
Algebraic number
Irregular
Necessary Conditions
Sufficient Conditions
Theorem

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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AB - The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a, q) at any distinct algebraic points to be algebraically independent, where Θ(x,a, q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a, q) taking algebraically independent values for any distinct triplets (x,a, q) of nonzero algebraic numbers. Moreover, Θ(a,a, q) is expressed as an irregular continued fraction and Θ(x, 1, q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.

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