Recounting Binomial Fibonacci Identities

Arthur T. Benjamin and Jeremy A. Rouse

A chapter in Applications of Fibonacci Numbers, 2004, pp 25-28 from Springer

Abstract: Abstract In [4], Carlitz demonstrates (1) $$ {F_L}\sum\limits_{{x_1} = 0}^n {\sum\limits_{{x_2} = 0}^n {...\sum\limits_{{x_{L = 0}}}^n {\left( \begin{gathered}n - {x_L} \hfill \\{x_1} \hfill \\\end{gathered} \right)} } } \left( \begin{gathered}n - {x_1} \hfill \\{x_2} \hfill \\\end{gathered} \right)...\left( \begin{gathered}n - {x_L} - 1 \hfill \\{x_L} \hfill \\\end{gathered} \right) = {F_{\left( {n + 1} \right)L,}} $$ using sophisticated matrix methods and Binet’s formula. Nevertheless, the presence of binomial coefficients suggests that an elementary combinatorial proof should be possible. In this paper, we present such a proof, leading to other Fibonacci identities.

Keywords: 05A19; 11B39 (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-306-48517-6_4

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