### 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 language | English |
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Pages (from-to) | 3081-3093 |

Number of pages | 13 |

Journal | Journal of Number Theory |

Volume | 129 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2009 Dec |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences.** / Tanaka, Takaaki.

Research output: Contribution to journal › Article

}

TY - JOUR

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

AU - Tanaka, Takaaki

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

UR - http://www.scopus.com/inward/record.url?scp=81755168769&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=81755168769&partnerID=8YFLogxK

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 -