2011/03/23 by Darij Grinberg, Grinberg, Darij
Mathematics · Physics and Astronomy · #05A19 #Advanced Mathematical Identities #Advanced Mathematical Theories #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1103.4507
openalex publication_date 2011/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Philip Matchett Wood and Doron Zeilberger have constructed identities for the Fibonacci numbers fn of the form 1fn = fn for all n ≥ 1; 2fn = fn-2 + fn+1 for all n ≥ 3; 3fn = fn-2 + fn+2 for all n ≥ 3; 4fn = fn-2 + fn + fn+2 for all n ≥ 3; ...; the general identity in this family has the form kfn = ∑s ∈ Sk fn+s (for all sufficiently high n), where Sk is a finite set of integers that depends only on k and contains no two consecutive integers. These identities are generalized, replacing the left-hand side kfn by arbitrary sums of the form fn+a1 + fn+a2 + ⋯ + fn+ap for arbitrary integers a1, a2, …, ap. The resulting theorem is proved using the connection between the Fibonacci numbers and the golden ratio.