Golden Ratio Base Expansions of the Logarithm and Inverse Tangent of Fibonacci and Lucas Numbers

2023/03/19 by Kunle Adegoke, Adegoke, Kunle, Jaume Oliver Lafont +1 · 1 citation
Mathematics · Physics and Astronomy · #11Y60 #30B99 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2303.13374

openalex publication_date 2023/03/19 · openalex created_date 2023/03/25 · openalex updated_date 2026/07/28

Abstract

Let α=(1+√ 5)/2, the golden ratio, and β=-1/α=(1 - √ 5)/2. Let F_n and L_n be the Fibonacci and Lucas numbers, defined by F_n=(αⁿ -βⁿ)/√ 5 and L_n=αⁿ + βⁿ, for all non-negative integers. We derive base~α expansions of log F_n, log L_n, \arctan\dfrac1F_n and \arctan\dfrac1L_n for all positive integers n.

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