### 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 language | English |
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Title of host publication | Diophantine Analysis and Related Fields, DARF 2007/2008 |

Pages | 190-204 |

Number of pages | 15 |

DOIs | |

Publication status | Published - 2008 Dec 1 |

Event | Diophantine Analysis and Related Fields, DARF 2007/2008 - Kyoto, Japan Duration: 2008 Mar 5 → 2008 Mar 7 |

### Publication series

Name | AIP Conference Proceedings |
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Volume | 976 |

ISSN (Print) | 0094-243X |

ISSN (Electronic) | 1551-7616 |

### Other

Other | Diophantine Analysis and Related Fields, DARF 2007/2008 |
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Country | Japan |

City | Kyoto |

Period | 08/3/5 → 08/3/7 |

### 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