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

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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.

元の言語English
ページ(範囲)3081-3093
ページ数13
ジャーナルJournal of Number Theory
129
発行部数12
DOI
出版物ステータスPublished - 2009 12

ASJC Scopus subject areas

  • Algebra and Number Theory

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