The hyperbolic-type k-Fibonacci sequences and their applications

Abstract

In this study, we define hyperbolic-type k-Fibonacci numbers and then give the relationships between the k-step Fibonacci numbers and the hyperbolic-type k-Fibonacci numbers. In addition, we study the hyperbolic-type k-Fibonacci sequence modulo m and then we give periods of the Hperbolic-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 hyperbolic-type k-Fibonacci sequences to groups. Finally, we obtain the periods of the hyperbolic-type 2-Fibonacci sequences in the dihedral group D2m, (m ? 2) with respect to the generating pairs (x,y) and (y, x).