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 with respect to the generating pairs (x, y) and (y, x).