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

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 $$ {F_z} = \left( {{\alpha ^z} - {\beta ^z}} \right)/\sqrt 5 $$ (1.1) $$ {L_z} = {\alpha ^z} + {\beta ^z}, $$ (1.2) where the subscript z is an arbitrary complex number and \( \alpha = - 1/\beta = \left( {1 + \sqrt 5 } \right)/2 \)