The Fibonacci sequence and an elliptic curve
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 483-495.

Infinite series involving the reciprocal Fibonacci numbers may admit no algebraic dependence between each other over the rational numbers. In this note, we introduce an identity which reveals an algebraic dependence relation between two infinite series involving the reciprocal Fibonacci numbers. The identity was discovered from a peculiar description of an elliptic function, and this observation is generalized to produce similar identities on a large class of sequences defined by linear recurrences on three consecutive terms.

Les séries infinies impliquant les inverses des nombres de Fibonacci sont en général algébriquement indépendantes sur le corps des nombres rationnels. Dans la présente note, nous introduisons une identité qui révèle une relation de dépendance algébrique entre deux telles séries. L’identité a été découverte à partir d’une description spéciale d’une certaine fonction elliptique. Cette observation est généralisée pour produire des identités analogues pour une grande classe de suites définies par des récurrences linéaires portant sur trois termes consécutifs.

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Published online:
DOI: 10.5802/jtnb.1210
Classification: 11B39, 11G05
Keywords: Fibonacci sequence, elliptic curves, $q$-series
Sungkon Chang 1

1 11935 Abercorn St, Savannah GA, 31419 United States
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Sungkon Chang. The Fibonacci sequence and an elliptic curve. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 483-495. doi : 10.5802/jtnb.1210. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1210/

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