k–step Fibonacci sequences and Fibonacci matrices

Abstract

In this paper, the powers of the Fibonacci matrices are investigated and closed formulas for their entries are derived, related to the suitable terms of the k− step Fibonacci sequences in order to develop the properties of the irreducibility and primitivity of the Fibonacci matrices. Some bounds for the spectral radius and the modulus of the remaining eigenvalues of the Fibonacci matrices are presented. New formulas for computing of the odd and even terms of a special Fibonacci sequence are discussed, generalizing the Cassin’s and Sharpe’s formulas.

Keywords:

• Fibonacci numbers and polynomials
• Spectral radius
• Irreducible
• Primitive

Mathematics Subject Classification:

• 11B39
• 15A18
• 15B48