The Ring of Fibonacci (Fibonacci “Numbers” with Matrix Subscript)

Odoardo Brugia, Piero Filipponi and Francesco Mazzarella

A chapter in Applications of Fibonacci Numbers, 1991, pp 51-62 from Springer

Abstract: Abstract Several authors (e.g., see [8]) have considered the Fibonacci numbers F x where the subscript x is an arbitrary real number and showed that these (complex) numbers enjoy most of the properties of the usual Fibonacci numbers F m (m integral). A quite natural extension of the numbers F x leads to the definition of the Fibonacci numbers F z and Lucas numbers L z 1.1 $$ {F_z} = \left( {{\alpha ^z} - {\beta ^z}} \right)/\sqrt 5 $$ 1.2 $$ {L_z} = {\alpha ^z} + {\beta ^z}, $$ where the subscript z is an arbitrary complex number and $$ \alpha = - 1/\beta = \left( {1 + \sqrt 5 } \right)/2 $$

Keywords: Real Matrix; Fibonacci Number; Arbitrary Real Number; Matrix Identity; Matrice Band (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-94-011-3586-3_7

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