### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Cellular and Molecular Neuroscience
- Clinical Neurology
- Molecular Biology

### Cite this

*Molecular Neurodegeneration*,

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

**Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers.** / Elsner, C.; Shimomura, S.; Shiokawa, I.

Research output: Contribution to journal › Article

*Molecular Neurodegeneration*, vol. 3, no. 1, pp. 429-446. https://doi.org/10.1007/s11139-007-9019-7

}

TY - JOUR

T1 - Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

AU - Elsner, C.

AU - Shimomura, S.

AU - Shiokawa, I.

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

KW - Algebraic independence

KW - Fibonacci numbers

KW - Jacobian elliptic functions

KW - Lucas numbers

KW - Nesterenko's theorem

KW - Q-series

KW - Ramanujan functions

UR - http://www.scopus.com/inward/record.url?scp=56549102668&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=56549102668&partnerID=8YFLogxK

U2 - 10.1007/s11139-007-9019-7

DO - 10.1007/s11139-007-9019-7

M3 - Article

AN - SCOPUS:56549102668

VL - 3

SP - 429

EP - 446

JO - Molecular Neurodegeneration

JF - Molecular Neurodegeneration

SN - 1750-1326

IS - 1

ER -