Abstract
In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ ∞ n=1 F −12n−1 , ∑ ∞ n=1 F −22n−1 , ∑ ∞ n=1 F −32n−1 and write each ∑ ∞ n=1 F −s2n−1 (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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References
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Elsner, C., Shimomura, S. & Shiokawa, I. Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers. Ramanujan J 17, 429–446 (2008). https://doi.org/10.1007/s11139-007-9019-7
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DOI: https://doi.org/10.1007/s11139-007-9019-7
Keywords
- Algebraic independence
- Fibonacci numbers
- Lucas numbers
- Jacobian elliptic functions
- Ramanujan functions
- q-series
- Nesterenko’s theorem