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

In: Applications of Fibonacci Numbers

Author

Listed:
  • Odoardo Brugia
  • Piero Filipponi
  • Francesco Mazzarella

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 $$ \alpha = - 1/\beta = \left( {1 + \sqrt 5 } \right)/2 $$

Suggested Citation

  • Odoardo Brugia & Piero Filipponi & Francesco Mazzarella, 1991. "The Ring of Fibonacci (Fibonacci “Numbers” with Matrix Subscript)," Springer Books, in: G. E. Bergum & A. N. Philippou & A. F. Horadam (ed.), Applications of Fibonacci Numbers, pages 51-62, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-3586-3_7
    DOI: 10.1007/978-94-011-3586-3_7
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