Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

C. Elsner; S. Shimomura; I. Shiokawa

doi:10.1007/s11139-007-9019-7

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

C. Elsner, S. Shimomura, I. Shiokawa

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

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
Pages (from-to)	429-446
Number of pages	18
Journal	Molecular Neurodegeneration
Volume	3
Issue number	1
DOIs	https://doi.org/10.1007/s11139-007-9019-7
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

Molecular Biology
Clinical Neurology
Cellular and Molecular Neuroscience

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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.",

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AU - Elsner, C.

AU - Shimomura, S.

AU - Shiokawa, I.

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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.

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KW - Fibonacci numbers

KW - Jacobian elliptic functions

KW - Lucas numbers

KW - Nesterenko's theorem

KW - Q-series

KW - Ramanujan functions

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Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

Abstract

Keywords

ASJC Scopus subject areas

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