The complex-type k-Fibonacci sequences and their applications

Abstract

In this article, we define the complex-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the complex-type k-Fibonacci numbers. Also, we obtain miscellaneous properties of the complex-type k-Fibonacci numbers such as the Binet formulas, the combinatorial, permanental, determinantal representations and the sums. In addition, we study the complex-type k-Fibonacci sequence modulo m and then we give some results concerning the periods and the ranks of the complex-type k-Fibonacci sequences for any k and m which are related the periods of the k-step Fibonacci sequences modulo m. Furthermore, we extend the complex-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the complex-type 2-Fibonacci sequences in the dihedral group $D_{2m},(m\ge 2)$ with respect to the generating pairs (x, y) and (y, x).

Keywords:

• The complex-type sequence
• group
• matrix
• period
• rank
• representation

2010 Mathematics Subject Classification:

• 11K31
• 39B32
• 15A15
• 11B50
• 11C20
• 05A15
• 20F05