### 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 | AIP Conference Proceedings |

Pages | 190-204 |

Number of pages | 15 |

Volume | 976 |

DOIs | |

Publication status | Published - 2008 |

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

### 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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*AIP Conference Proceedings*(Vol. 976, pp. 190-204) https://doi.org/10.1063/1.2841905

**Remarkable algebraic independence property of certain series related to continued fractions.** / Tanaka, Takaaki.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*AIP Conference Proceedings.*vol. 976, pp. 190-204, Diophantine Analysis and Related Fields, DARF 2007/2008, Kyoto, Japan, 08/3/5. https://doi.org/10.1063/1.2841905

}

TY - GEN

T1 - Remarkable algebraic independence property of certain series related to continued fractions

AU - Tanaka, Takaaki

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Algebraic independence

KW - Continued fractions

KW - Fibonacci numbers

KW - Mahler's method

UR - http://www.scopus.com/inward/record.url?scp=77958172298&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77958172298&partnerID=8YFLogxK

U2 - 10.1063/1.2841905

DO - 10.1063/1.2841905

M3 - Conference contribution

AN - SCOPUS:77958172298

SN - 9780735404953

VL - 976

SP - 190

EP - 204

BT - AIP Conference Proceedings

ER -