TY - JOUR

T1 - Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

AU - Tanaka, Taka aki

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2009/12

Y1 - 2009/12

N2 - 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.

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.

KW - Algebraic independence

KW - Continued fractions

KW - Fibonacci numbers

KW - Mahler's method

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U2 - 10.1016/j.jnt.2009.06.004

DO - 10.1016/j.jnt.2009.06.004

M3 - Article

AN - SCOPUS:81755168769

VL - 129

SP - 3081

EP - 3093

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 12

ER -