Generalizations of a Fibonacci Identity

F. T. Howard

A chapter in Applications of Fibonacci Numbers, 1999, pp 201-211 from Springer

Abstract: Abstract A well-known identity is $$F_{m + n} = L_mF_n + (-1)^{m-1}F_{n-m}$$ where F k and L k are the Fibonacci and Lucas numbers, respectively. With the definitions $$F_{-k} = (-1)^{k + 1}F_k$$ and $$L_{-k} = (-1)^kL_k$$ formula (1.1) is true for all integer m and n. The identity is easy to prove, and it is evidently useful; Rokach [11], for example, proved that (1.1) is about 2.88 times more efficient than the usual Fibonacci recurrence for computing the Fibonacci numbers.

Keywords: 11B39; 11B37 (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-94-011-4271-7_20

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