### Abstract

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _{n=1} ^{∞} F _{2n-1} ^{-1} , ∑ _{n=1} ^{∞} F _{2n-1} ^{-2} , ∑ _{n=1} ^{∞} F _{2n-1} ^{-3} and write each ∑ _{n=1} ^{∞} F _{2n-1} ^{-s} (s>4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.

Original language | English |
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Pages (from-to) | 429-446 |

Number of pages | 18 |

Journal | Molecular Neurodegeneration |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Nov 27 |

### Keywords

- Algebraic independence
- Fibonacci numbers
- Jacobian elliptic functions
- Lucas numbers
- Nesterenko's theorem
- Q-series
- Ramanujan functions

### ASJC Scopus subject areas

- Molecular Biology
- Clinical Neurology
- Cellular and Molecular Neuroscience

## Cite this

Elsner, C., Shimomura, S., & Shiokawa, I. (2008). Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers.

*Molecular Neurodegeneration*,*3*(1), 429-446. https://doi.org/10.1007/s11139-007-9019-7